het mtUuuraitty of himago THE CLASS NUMBER OF BINARY QUADRATIC FORMS A DISSERTATION SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF SCIENCE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY GEORGE HOFFMAN CRESSE Reprinted from DICKSON'S HISTORY OF THE THEORY OF NUMBERS, VOL. III, CH. VI THE CARNEGIE INSTITUTION OF WASHINGTON 1923 CHAPTER VI. NUMBER OF CLASSES OF BINARY QUADRATIC FORMS WITH INTEGRAL COEFFICIENTS. INTRODUCTION. Particular interest in the mere number of the classes of binary quadratic forms of a given determinant dates from the establishment by C. F. Gauss of the relation between the number h of properly primitive classes of the negative determinant -D and the number of proper representations of D as the sum of three squares. Gauss himself found various expressions for h. G. L. Dirichlet elaborated Gauss' method exhaustively and rigorously. L. Kronecker, by a study of elliptic modular equations, deduced recurrence formulas for class-number which have come to be called class-number relations. C. Hermite obtained many relations of the same general type by equating certain coefficients in two different expansions of pseudo-doubly periodic functions. Hermite's method was extended by K. Petr and G. Humbert to deduce all of Kronecker's relations as well as new and independent ones of the same general type. The method of Hermite was translated by J. Liouville into a purely arithmetical deduction of Kronecker's relations. The modular function of F. Klein, which is invariant only under a certain congruencial sub-group of the group of unitary substitutions, was employed by A. Hurwitz and J. Gierster just as elliptic moduli had been employed by Kronecker and so the range of class-number relations was vastly extended. Taking the suggestion from R. Dtdekind in his investigation of the classes of ideals of the quadratic field of discriminant D, Kronecker departed from the tradition of Gauss and chose the representative form ax2+ bxy + cy2, where b is indifferently odd or even, and regarded as primitive only forms in which the coefficients have no common divisor. Kronecker thus simplified Dirichlet's results and at the same time set up a relation in terms of elliptic theta functions between the classnumber of two discriminants; so he referred the problem of the class-number of a positive discriminant to that of a negative discriminant. By a study of quadratic residues, M. Lerch and others have curtailed the computation of the class-number. A. Hurwitz has accomplished the same object by approximating h(p), p a prime, by a rapidly converging series and then applying a congruencial condition which selects the exact value of h (p). Reports are made on several independent methods of obtaining the asymptotic expression for the class-number, and also methods of obtaining the ratio between 92 CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 93 the number of classes of different orders of the same determinant. The chief advances that have been made in recent years have been made by extending the method of Hermite. We shall frequently avoid the explanation of an author's peculiar symbols by using the more current notation. Where there is no local indication to the contrary, h (D) denotes the number of properly primitive, and h'(D) the number of improperly primitive, classes of Gauss forms (a, b, c) of determinant D= b2-ac. Referring to Gauss' forms, F(D), G(D), E(D), though printed in italics, will have the meaning which L. Kronecker (p. 109) assigned to them when printed in Roman type. The class-number symbol H(D) is defined as G(D) -F(D). By K(D) or CID, we denote the number of classes of primitive Kroneeker forms of discriminant D-b2-4ac. A determinant is fundamental if it is of the form P or 2P; a discriminant is fundamental if it is of the form P, 4P=4(4n-1) or 8P, where P is an odd number without a square divisor other than 1. The context will usually be depended on to show to what extent the Legendre symbol (P/Q) is generalized. Reduced form and equivalence will have the meanings assigned by Gauss (cf. Ch. I). Among definite forms, only positive forms will be considered; and the' leading coefficient of representative indefinite forms will be understood to be positive. Ordinarily, r will be used to denote the number of automorphs for a form under consideration; but when D>0, T=1. Some account will be given of the modular equations which lead to class-number relations. In reports of papers involving elliptic theta functions, the notations of the original authors will be adopted without giving definitions of the symbols. For the definitions and a comparison of the systems of theta-function notation, the reader is referred to the accompanying table. The different functions of the divisors of a number will be denoted by the symbols of Kronecker,54 and without repeating the definition. A Gauss form will be called odd if it has at least one odd outer coefficient; otherwise it is an even form. These terms are not applied to Kronecker forms. TABLES OF THETA-FUNCTIONS. CD o1 ~ oor 1 a | e (z) or e-So(a)=S(x, r) =o00()= oo()=,02(v)=e( (x) or e =s,(x)=-o,(v, q) ~1(z) or lH=3(x)=S2(x,r)=lo(z)-=Olo(v) — 1(v)=i,(x)o r =1 32(Z)=2(v, q) H (z) or = -(x)( —(c,r)=0l(z)=Oll(v)= L (v)ff (sx) or H =1(x)= (v, q) Here, r=q2, z=2Kx/7r, V-=x/7r, n is any, m is any odd, integer; and, according to Humbert, 00 00 e~(x) =-q2 cos 2x, @(x) =(-])Vq2 cos 2nx, H (x) =qm24 cos mx, H(X)=S(-l)(-1') 1)o/4 -sin mx. n=-oo n==-oo 94 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI For x= 0, the following systems of special symbols are represented in this chapter. 01 ( (q)or -(q) 000=01=01=3= (0) =,io (q) or o (q) = S= 0 = -0 2=4 = t ~ (o)=.o (q) or o (q)=Ooi-0 =0 -e~=94='Hf,(0) =2 (q) or 02(q) =o01 = = 1=1 02~ = H (O) = /(q) or 01'(q)=7r0el' = 79 =-' In connection with these tables, the following relations will need to be recalled: q ei, =r-TiK'/K; VK = 02(q)/0,(q), VK-0(q)/03 (g), 01(q) = (q)02 q (q) ); V2K/7=03(q), VK'K/r=0(q), V2KK/7T=02(q); A. M. Legendrel excluded every reduced form ("quadratic divisor") whose determinant has a square divisor. Each reduced form py2+2qyz+ 2mz2 of determinant -a= - (4n+ 1) has a conjugate reduced form 2py2+ 2qyz+ mz2; here p, q, m are all odd. If a is of the form 8n+ 5, one of p, m is of the form 4n+1 and the other of the form 4n -1. Hence the odd numbers represented by one of the quadratic forms are all of the form 4n+ 1 and those represented by the conjugate form are of the form 4n+ 3. Thus a form and its conjugate are not equivalent and the total number of reduced forms is even. If a=8n+l, the number of reduced forms may be even or odd,1 but is odd2 if a= 8n +1 is prime. Legendre3 counted (r, s, t) and (r, -s, t) as the same form. Hence for a= 4n+ 1, his number of forms is f h7(-a) -A \, where h (-a), in the terminology of Gauss4 (Art. 172), is the number of properly primitive classes and A is the number of ambiguous properly primitive classes plus the number of classes represented by forms of the type (r, s, r). C. F. Gauss,4 by the composition of classes, proved (Art. 252) that the different genera of the same order have the same number of classes (cf. Ch. IV). He5 then set for himself the problem of finding an expression in terms of D for the number of classes in the principal genus of determinant D. He succeeded later8 in finding an expression for the total number of primitive classes of the determinant and thus solved his former problem only incidentally. 1Theorie des nombres, Paris, 1798, 267-8; ed. 2, 1808, 245-6; ed. 3, 1830, Vol. I, Part II, ~ XI (No. 217), pp. 287-8; German transl. by H. Maser, Zahlentheorie, I, 283. 2 Ibid., Part IV, Prop. VIII, 1798, 449; ed. 2, 1808, 385; ed. 3, II, 1830, 55; Zahlentheorie, II, 56. 3 Ibid., 1798, No. 48, p. 74; ed. 2, 1808, p. 65; ed. 3, I, p. 77; Zahlentheorie, I, p. 79. 4 Disquisitiones Arithmeticae, 1801; Werke, I, 1876; German transl. by H. Maser, Untersuchungen ueber Hohere Arithmetik, 1889; French transl. by A. C. M. Poullet-Delisle, Reserches Arithmetiques, 1807, 1910. 5 Werke, I, 466; Maser, 450; Supplement X to Art. 306. Cf. opening of Gauss' 8, memoirs of 1834, 1837. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 95 If (Art. 253) Q denotes the number of classes of the (positive) order 0 of determinant D, and if r denotes the number of properly primitive classes of determinant D which, being compounded with an arbitrary class K of the order 0, produce a given arbitrary class L of the order 0, then the number of properly primitive (positive) classes is rQ. We take both K and L to be the simplest form (Art. 250). It is proved (Arts. 254-6) by the composition of forms that the above r classes are included among certain r' primitive forms, r' being given by r -doa L \a/ ay in which (A, B, C) is the simplest form of order 0, D'=4D/A2, and a ranges over the distinct odd divisors of A, while n=2 if D/A2 is an integer, n= 1 if 4D/A2 1 (mod 8), n=3 if 4D/A2 5 (mod 8). Now r=r' if D is a positive square or a negative number except in the cases D= -A2 and -4A2, in which cases r=r'/2 and r'/3 respectively. No general relation (Art. 256, IV, V) is found between r and r' for D positive and not a square. The problem of finding the ratio of the number of classes of different orders of a determinant will be hereafter referred to as the Gauss Problem. It was solved completely by Dirichlet,20'93 Lipschitz,41 Dedekind,115 Pepin,120'137 Dedekind,127' Kronecker,171 Weber,220 Mertens,237 Lerch,277 Chatelain,316 and de Seguier.226 If 0 is the improperly primitive order, the same method gives the following result (Art. 256, VI): If D_1 (mod 8), r=l; if D<0 and =5 (mod 8), r=3 (except when D= -3 and then r=l); if D>0 and =5 (mod 8), r=l or 3, according as the three properly primitive forms (1, 0, -D)>, (4, 1, 1(1-D)), (4, 3, 1(9-D)) belong to one or three different classes. Gauss (Art. 302) gave the following expression for the asymptotic median number of the properly primitive classes of a negative determinant -D: M(D) =mVD —, 7(1+ + ++..)' He later corrected6 this formula to m VD. His tables of genera and classes led him (Art. 303) to the conjecture304 that the number of negative determinants which have a given class-number h is finite for every h (cf. Joubert,60 Landau,260 Lerch,262 Dickson,327 Rabinovitch336' and Nagel3361). The asymptotic median value of h(k2) is 8k/7r2 (Art. 304). He conjectured that the number of positive determinants which have genera of a single class is infinite. Dirichlet40 proved that this is true. He stated (Art. 304) that, for a positive determinant D, the asymptotic median value of h (D) log(T+ U VD) is mn/ D-n, where T, U give the fundamental solution of t2-Du'= 1 and7 for m as above, while n is a constant as yet not evaluated (cf. Lipschitz102). 6 Werke, II, 1876, 284; Maser's transl., 670. Cf. Lipschitz.l02 7 On the value of m, see Supplement referring to Art. 306 (X). Maser's transl., p. 450; Werke, 1, 1863, 466. 96 HISTORY OF THE THEORY OF NUM:BERS. [CHAP. VI C. F. Gauss8 considered the lattice points within or on the boundary of an ellipse ax2-+2bxy+cy2=A, where A is a positive integer. The area is rA/VD, where -D=b2-ac. Hence as A increases indefinitely, the number of representations of all positive numbers ~ A by means of the definite form (a, b, c) bears to A a ratio which approaches rr/V/D as a limit. Hereafter9 the determinant - D has no square divisors, and the asymptotic number of representations of odd numbers - M by the complex C of representative properly primitive forms of determinant -D is - h(-D). 2XVD To evaluate h (-D), a second expression for this number of representations is found; but Gauss gives the deduction only in fragments. Thus if (n) denotes the number of representations of n by C and p is an odd prime, then10 1. (pn) = (n), if p is a divisor of D; 2. (n)-(n) + (h), if (; 3. (pn)=-(n)+h,if ( )= — where n = hp/, [u arbitrary, h prime to p. This implies in the three cases 1. (h) =(ph)=(ph)= (p3h)=...; 2. (ph)=2(h), (p2h)=3(h), (ph)=4(h),...; 3. (p)= =0, (p2h)=(h), (p3h)=0, (ph) =(h),.... Hence the ratio of the mean number of representations by C of all odd numbers <1 M to the mean number of representations of those numbers after the highest possible power of p has been removed from each as a factor is, in each of the three cases,'1 A second odd prime divisor p' is similarly eliminated from the odd numbers M;; and so on. Eventually the number of representations of the numbers is asymptotically ~rM. Gauss, supposing -D<-1, takes the number r of automorphs to be 2. (See Disq. Arith., Art. 179; Gauss35 of Ch. I.) Hence the original number of representations is asymptotically12 M/i (1 ( D ) 1) 8Posthumous paper presented to Konig. Gesells. der Wiss. Gottingen, 1834; Werke, II, 1876, 269-276; Untersuchungen iiber Hohere Arith., 1889, 655-661. 9 Posthumous fragmentary paper presented to Konig. Gesells. der Wiss. Gothingen, 1837; Werke, II, 1876, 276-291; Untersuchungen liber Hohere Arith., 1889, 662-677. 10 Cf. remarks by R. Dedekind, Werke of Gauss, 1876, II, 293-294; Untersuchungen iiber Hihere Arith., 1889, 686. 11 Cf. R. Dedekind, Werke of Gauss, II, 1876, 295-296; Untersuchungen, 1889, 687. 12 Cf. remarks of R. Dedekind, Werke of Gauss, II, 1876, 296; Untersuchungen, 688. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 97 And hence (Untersuchungen, 670, III; cf. Dirichlet,19 (1)) h(-D) _ { - p ) p Gauss gives without proof five further forms of h(-D) including h(-D) =: V D ) cot nO, where 0=r/N, N =D or 4D, n is odd > 0 and <D. Cf. Lebesgue,36 (1). By considering the number of lattice points in a certain hyperbolic sector,3 h (D) is found to be, for D>0, 0 30 50 2VD(l~-~_~...) log sin log sin- ~logsin- ~.. log(T+U VD) - log(T+U /D) where the coefficient ~+1 of 1/in and of log sin mO/2 is (D/m). Cf. Dirichlet,23 (7), (8). For a negative prime determinant, -D=- (4n +1), h (-D) is stated incorrectly to be a-/3, where a and P/ are respectively the number of quadratic residues and nonresidues of D in the first quadrant of D. [This should be 2(a-/3); see Dirichlet,23 formula (5).] Extensive tables lead by induction to laws which state, in terms of the classnumber of a prime determinant p, the distribution of quadratic residues of p in its octants and 12th intervals. G. L. Dirichlet14 obtained h( - q), where q is a positive prime =4n+ 3>3. By replacing infinite sums by infinite products he obtained the lemma: L m \ q / ns / 2 where n ranges in order over all positive odd integers prime to q, and m ranges over all positive numbers which have only prime divisors f such that (f/q) = 1; while KJ is the number of such distinct divisors of m; and s is arbitrary >1. Now ax2 + 2bxy + cy2, ax2+ 2b'xy + cy2,. denotes a complete set of representative properly primitive (positive) forms of determinant -q. Then, by the lemma, since the number of representations of m by the forms is 2P+1 (cf. Dirichlet, Zahlentheorie, ~ 87), we have (1) 2 ns q/S n ~ (ax2+2bxy+cy2)8 ~+ (a'x2 +2by+c'y 2)8 ' where x, y take every pair of values for which the values of the quadratic forms are 13 Remarks of R. Dedekind, Gauss' Werke, II, 1876, 299; Maser's translation, 691. Cf. G. L. Dirichlet, Zahlentheorie, ~98. 4 Jour. fur Math., 18, 1838, 259-274; Werke, I, 1889, 357-370. 98 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI prime'5 to 2q. We let s1 + p, and let p>O approach zero. The limit of the ratio of each double sum in the right member to q- 7r 2qVq p is found from the lattice points of an ellipse to be 1. But lim:~I q-1 1 lim n - H:.- =. ns 2q p Hence (cf. C. F. Gauss,9 Werke, II, 1876, 285), (2) h(-q)=2V () S. Tr n=lo f n 7 To evaluate S, we consider 1 2 = (n s n l where n now ranges in order over all integers - 1. In the cyclotomic theory,16 Esin 2a sin 2bn7r ( n \ q q \q Hence Vq< o 1. 2anr 1 l 2hnr -/' -. S=- - sin -- -S i - sin - /2 JL a n q q b n n q \q 2 where (a/q)=1, (b/q)= -1, and a, b are >0 and <q. Since (cf. W. E. Byerly, Fourier's Series, 1893, 39) z=2b7r/q is between 0 and 2ir, -= - sin nz; 2 and so17 (3) =2 [ 2 1 5b aEvaluating S itself by cyclotomic considerations, Dirichlet gives the result's (4) h=A-B=2A - (q- 1), where A and B are respectively the number of quadratic residues and non-residues of q which are < q. For p=4n + 1, Dirichlet obtained (5) -h(-p) =2(A-B) =4A- (p- ) 15 This restriction is removed by G. Humbert, Comptes Rendus, Paris, 169, 1919, 360-361. 16 Cf. C. F. Gauss, Werke, II, 1876, 12. G. L. Dirichlet, Zahlentheorie, ~116. 17 Stated empirically by C. G. J. Jacobi, Jour. fur Math., 9, 1832, 189-192; detailed report in this History, Vol. I, 275-6; J. V. Pexider,320 Archiv Math. Phys., (3), 14, 1909, 84-88, combined (3) with the known relation Zb -+ a- == q (q -1) to express h in terms of 2a alone or 2b alone. 18 G. B. Mathews, Proc. London Math. Soc., 31, 1899, 355-8, expressed A - B in terms of the greatest integer function. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 99 where A and B are the number of a's and b's respectively between 0 and 3p; and without proof, he stated that h(-pq) = (b-:a)/pq or 3 (b-a) /pq, according as pq is -7 or 3 (mod 8), where a, b are positive integers <pq, and (a/p) = (a/q), (b/p) - (b/q). For h(q), the factor 7r in (2) must be replaced by log(T+UVD); as the lattice points involved must now lie in a certain hyperbolic sector rather than an ellipse (cf. Gauss,9 Dirichletl9). G. L. Dirichlet19 considered the four cases of a determinant: D=P.S2, P= 1 and 3 (mod 4); D=2P* S, P 1 and 3 (mod 4), where S2 is the greatest square divisor of D. He defined 8 and E in the four cases as follows: 8=E=l, — 1, = —, -e, =1,=-1, 8=e= ---1. Employing the notation of his former memoir,14 he found for all four cases, if m is representable, 8(f-_l)E(f-2_) (_f) = 1 Consequently the generalization of (1) of the preceding memoir14 is, for D= - D <0, 1 -2___ 12 1 C ty2_ +. =__ 41 _4-1)j^n-2_- 3 (ax'2 — bxy + cy') s (a'x2 + 2bxy +cy') 8 -. =2 1 () P e n) where the restrictions on s, x, y, n are the same as for (1) in the preceding memoir. A lemma shows that 1 (D,) 1 r(D,) 1 ]: — D - -.or —. n~ 2D,1 p D1 P ' according as D is odd or even, where s=1 +p and p is indefinitely small, and p is the Euler symbol. The study of lattice points in the ellipse ax2 + 2bxy + cy2 =N for very great N leads to 7r +(D,) 1 7r(Dl) 1 - - or 2 VD1 p D p as the asymptotic value of each of the h sums in the first member, according as D is odd or even. Hence for D= -D <0, (1) h= 2 h/D-8~("-1)("2-l) (n )1 7r \P n Dirichlet obtained independently an analogous formula for the number h' of improperly primitive classes of determinant D =- D <O. For D> 0, results analogous to all those for D<0, are obtained by considering all the representations of positive numbers < N by ax2+2bxy +cy2, where a is >0 and (x, y) are lattice points in the hyperbolic sector having y>0 and bounded by y 0, U(ax+- by) =Ty, and ax2 +2bxy + cy2=N. For D>0, these restrictions on a, x, y are hereafter understood in this chapter of the History. 9 Jour. fur Math., 19, 1839, 324-369; 21, 1840, 1-12, 134-155; Werke, I, 1889, 411-496; Ostwald's Klassiker der exakten Wissenschaften, No. 91, 1897, with explanatory notes by R. Haussner. 100 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Incidentally Dirichlet~2 stated for D<-3 the "fundamental equation of Dirichlet" (see Zahlentheorie, ~ 92, for the general statement): (2):'( (ax2 2bxy - cy2) + S' (a'x2+2b'xy+c'y") +.. = 28s(n-1)(n2) (n) (n'), where p is an arbitrary function which gives absolute convergence in both members; the forms are a representative primitive system; x and y take all pairs of integral values (excepting x=y=0) in each form for which the value of the form is prime21 to 2D if the form is properly primitive, but half of the value of the form is prime to 2D if the form is improperly primitive; the second member is a double sum as to n and n'. Kronecker171 and Lerch277 (Chapter I of his Prize Essay) used this identity to obtain a class-number formula. Dirichlet noted from the results in his19 former memoir that for D<0, h=-h' or 3h', according as D=1 or 5 (mod 8), except that h=h' for D=-3. For D>0, if D= 8n + 1, h =h'; but, if D= 8n+ 5, h = h' or 3h', according as the fundamental solutions of t2 -Du2=4 are odd or even. (Cf. Gauss,4 Disq. Arith., Art. 256.) Since the series in (1) may be written as iI{i-8(n-1) c(n, -) (p) -8 where n is a positive odd prime, and prime to D, it follows that if h and h' denote respectively the number of properly primitive classes of the two negative determinants D and D'=D.S2, D having no square divisor, then h' -=SI - 18 2 8 ( ) ] where22 r ranges over the odd prime positive divisors of S (except if D=-1, the ratio thus given should be divided by 2). The corresponding ratio is found for D'>O. Dirichlet23 hereafter took S=1 and, representing the series in (1) by V, found that for D= - -P-1 (mod 4), for example, ___ __ - -d - e p -P 2mri dX nm- 1_ I 2.., P —1; P=mn o x n., m= 2, 1,,., P-1; P=D.tr. 2o Jour. fuir Math., 21, 1840, 7; Werke, I, 1889, 467. The text is a report of Jour. fuir Math. 21, 1840, 1-12; Werke, I, 1889, 461-72. 21 This restriction is removed by G. Humbert, Comptes Rendus, Paris, 169, 1919, 360-361. 22 Cf. Disq. Arith., Art. 256, V; R. Lipschitz,4' Jour. fiir Math., 53, 1857, 238. 23 From this point, Jour fir Math., 21, 1840, 134-155; Werke, I, 1889, 479-496. Cf. Zahlentheorie, ~~-103-105. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 101 The identity, in Gauss sums, (3) n_ n2mwi (P.- )2 (3) 2- Vi( P now gives ~ i/~ 2~) P m \ i i -_ ( -----: (- )(log sin mr - 7 1 1 V/P m —1 P P P whence V - - 2/P: >(ip)log sin mp, if P-=4 + 1, { V-= 3( -( -pi ( )m, if P=4t-l. For D= -P, P=4,A- 1, the comparison of (1) and (3) gives \ 2 / m\,1. 2mwr. h (DZ)) =-> ( p ) > ff sin n p ^W=^ [-jr)^^^ P n whence24 finally by grouping quadratic residues and non-residues, we have: So Dirichlet25 obtained his classic formulas for D0<O: D -P- P 4p+3, h(D)= D=-P, P=4-+ 1, h(D) =-( ); (5) D=-2P, P=4oJr+3, h(D)=2 (); D=-2P, P= 4t+, (D) = 2- - ( ) From (1) and (4) and their analogues, he wrote also in the four cases of D<O 0 (6) -h(D)=p (2-(-p) S- f ls1 4p S ( 2 (-p) -S S p }S' where S-=m ranges from 0 to P, 4P, 8P, 8P in the four respective cases, and 1=1, 2= (-l)iJ(-1), 3=(-)(i-1), e- =(_-)I(-1)+(m-l,). For D> >0, the analogue of (1) is ("*> '(D ) _" I _ lo( (TD+ U- ) o( P n 24 Fourier-Freeman, Theory of Heat, Cambridge, 1878, 243. 25 Jour. fir Math., 21, 1840, 152; Werke, I, 492-3; Zahlentheorie, ~106. 102 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI where T, U are the fundamental solution of t2 -Du2 1. Hence26 from equations like (4), we obtain: (81) D=P, P=4/+1, h(D)= lg( 1 lsin br/P log(T+ TUVP) 11 sin ar/P' where a and b range over the integers <P and prime to P for which (a/P)= +, (b/P)=-1; 1 IT sin I b /P (82) D=P, P=41 +3, h(D)= l og (T + sin ar/ log(T+ UVP) gI sin lair/P where a and b range over the integers m<4P and prime to 4P for which (- 1)(m-x) ( =p )-+ 1 or -1, according as m=a or b; (83) D=2P, h()= (T+ )log Hsin br/P log (T' + U /2P) II sin -lCa-/P' where a and b range over the integers m < 8P and prime to 8P, for which if P 1 (mod 4), (-1)(2-1) (P)=-+ or -1, according as m-a or b; if P 3 (mod 4), (-1) (-L)+(M2-l) (p) + 1 or -1, according as m=a or b. If D=P=44/+ 1>0, (4) and (7) with cyclotomic considerations give27 (9) h(D)=-[4-2 2) 2 og(+VP) where i[Y(x) + Z(x) VP] = II (x- e2:bi/P). Arndt53 supplied formulas for the other three cases. A. L. Cauchy28 proved that if p is a prime of the form 41+ 3, A-B 2 -3B(p+)/4 or B(P+i)/4 (mod p), according as p=81+ 3 or 8+ 7, where A is the number of quadratic residues and B that of the non-residues of p which are >0 and <-p, and Bk is the kth Bernoullian number. This implies, by G. L. Dirichlet,23 (5), that (1) h(-p) =2B(,+1)/4 or -6B(p,1)/4 (mod p), according as p=81+ 7 or 81+3 [cf. Friedmann and Tamarkine32T]. Cauchy29 obtained also the equivalent of the following for n free from square factors, and of the form 4x + 3: (2) A-B= [2-()b - a [2()12b2 (2) / ( iJ n i \( / ni n 26 Jour. fur Math., 21, 1840, 151; Werke, I, 492. 27 See this History, Vol. II, Ch. XII, 372 117; Cf. Dirichlet, Zahlentheorie, 1894, 279, ~ 107. 28 Mem. Institut de France, 17, 1840, 445; Oeuvres, (1), III, 172. Bull. Sc. Math., Phys., Chim. (ed., Ferussac), 1831. 29 Mem. Institut de France, 17, 1840, 697; Oeuvres, (1), III, 388. Comptes Rendus, Paris, 10, 1840, 451. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 103 where A, B are the number of quadratic residues and non-residues of n, which are <in, while a, b are >0 and <n, (a/n) =1, (b/n) = -1: and similar formulas for n=4x+l. Hence, for n=4x+3, h(-n)= _2- 2 52 -a2 n \^/J n2 is called Cauchy's class-number formula.30 M. A. Stern31 found that when P is a prime Smn+7, or 8m +3 respectively, II cot =~( —1)N a P VP where a ranges over all positive integers <P prime to P such that (a/P)= 1, and N denotes the number of quadratic divisors of determinant -P. This formula has been made to include the case P=4m+1 by Lerch.323 G. Eisenstein32 proposed the problem: If D>0 is S5 (mod 8), to determine a priori whether p2 - Dq2 =4 can be solved in odd or even integers p, q; that is33 to furnish a criterion to determine whether the number of properly primitive classes of determinant D is 1 or 3 times the number of improperly primitive classes of the same determinant. He also proposed the problem34: To find a criterion to determine whether the number of properly primitive classes of a determinant D is divisible by 3; and if this is the case, a criterion to determine those classes which can be obtained by triplication35 of other classes. V. A. Lebesgue36 employed the notation of Dirichlet23 and, in his four cases, set p=P, 4P, 8P, 8P, and f(x) =S e (a/p) x, summed over all the positive integers a<p, for i= 1, 2, 3, 4. Then v'- fI f(x)dx Jo.(l-p) is the sum of integrals (for the various values of a), with proper signs prefixed, la-ldx 1 P 2acr. r O ar Jo =1 - - 4cos -- log sn s + cot1-xp P M=l p P 2p p For a negative determinant, the terms involving the logarithm cancel each other and then, by the theory of Gauss37 sums, V' reduces to38 (1) Y'= 2r cot7t (, E )1, 1<A<p. H. W. Erler39 developed a hint by Gauss (Disq. Arith., Art. 256, ~ V, third case) that there is a remarkable relation between the totality B of properly primitive forms 30 Cf. T. Pepin,l20 Annales sc. de l'Ecole Norm. Sup., (2), 3, 1874, 205; M. Lerch,277 Acta Math., 29, 1905, 381. 31 Jour. de Math., (1), 5, 1840, 216-7. This is proved by means of C. G. J. Jacobi's result in this History, Vol. I, 275-6. 32 Jour. fir Math., 27, 1844, 86. 33 Cf. G. L. Dirichlet, Zahlentheorie, 1894, Art. 99; Dirichlet.23 84 Jour. fir Math., 27, 1844, 87. 35 Cf. C. F. Gauss, Disq. Arith., Art. 249; Maser's translation, 1889, 261; Werke, I, 1876, 272. 36 Jour. de Math., 15, 1850, 227-232. 37 Disq. Arith., Art. 356. 38 Cf. C. F. Gauss, memoir of 1837, Werke, II, 1876, 286; Untersuchungen, 1889, 671. 39 Eine Zahlentheoretische Abhandlung, Progr. Zullichau, 1855, p. 18. 104 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI of determinant D which represent A2 and the least solution tr, u of t2-Du2=A2. Erler considered the case in which A2 divides D, whence A divides t,. Write - = t/A, D'=D/A2, whence t2-D'u2 =1. Find the period of the solution of the latter for modulus A. From each pair of simultaneous values of r1, ua, we can derive one and only one from the set B which is equivalent to the principal form. The terms of every later period give the same forms in the same sequence as those of the first period. In case bisection of the period is possible, the terms of the second half are the same as in the first half. The forms obtained from the terms of the first half (or from the entire first period, if bisection is impossible) are distinct. G. L. Dirichlet40 recalled (see Dirichlet,23 (8)) that for a positive determinant D' =DS2, hh(D') =h(T+UD) log(TS.R=U, h(D).S.R log(T'+ U'VD') N in which R is independent of at, a2,..., ak in S=pL. pa2... p.. where the p's are distinct primes. By the theory of the Pell equation it is found (see this History, Vol. II, p. 377, Dirichlett34) that if each a increases indefinitely, S/N is eventually a constant. Hence for every D, there is an infinitude of determinants D'=DS2 for which h (D) =h (D). And a proper choice of D and the primes p, P2,..., pk leads to an infinite sequence of determinants D' for which the number of genera coincides with the value of h(D'). This establishes the conjecture of C. F. Gauss (Disq. Arith.,4 Art. 304) that there is an infinitude of determinants which have genera of a single class. R. Lipschitz41 called the linear substitutions /a, \ /A B Y(,;I )r A( equivalent if a,..., A are integers and if integers a, /3', yt, 8t exist such that (a a/'(a /t-/A B) 08B /-IL -/ ~ /I 8 r A; a - g -1I Every substitution of odd prime order p is equivalent to one of the p +1 non-equivalent substitutions: o1I 0 \ /O -p (p-1 p \1 O/' 0 r ' \ 1. OI Let (a, b, c), a properly primitive form of determinant D, be transformed by (1) into p+1 forms (a', b', c'). Then D'=D.p2. The coefficients of every form (a', b', c') satisfy the system of equations ap2_ =ca'82 _ 2 b'8y + C'y2, bp2 = - a'83 + b (a + py) - cya, cp2 = a'/2 - 2b'/a + C'a2. 40 Bericht. Acad. Berlin, 1855, 493-495; Jour. de Math., (2), 1, 1856, 76-79; Jour. fur Math., 53, 1857, 127-129; Werke, II, 191-194. 41 Jour. fiir Math., 53, 1857, 238-259. See H. J. S. Smith, Report Brit. Assoc., 1862, ~ 113; Coll. Math. Papers, I, 246-9; also G. B. Mathews, Theory of Numbers, 1892, 159-170. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 105 Hence (a', b', c') has no other divisor than p, and the condition that p be a divisor is aa'=a(aa2+2bay+cy2)= (aa +by)2-Dy2-0 (mod p). Now, a may be assumed relatively prime to p. The number of solutions of this congruence is the number of substitutions in (1) which do not lead to properly primitive forms (a', b', c'). This number is 2, 0, 1 according as (D/p) =1, -1, 0. Hence the number of properly primitive forms (a',,c') is p- (D/p). If (a I) be one substitution of (1) which carries (a, b, c) over into a particular (a', b', c'), then all the substitutions in (1) which effect the transformation are (A B), in which (Gauss, Disq. Arith., Art. 162; report in Ch. I) A=at- (ab +yc)u, r=yt+ (aa+yb)u, B=/3t- (/b +8c)u, A= St + (3a+Sb)u, t, u ranging over o pairs of integers which satisfy t2 - Du2= 1, where o is the smallest value of i for which ui is a multiple of p in (T+ UVD) =t +uiVD. If ua = pu', t = t', then (2T+ UVD) '=-t' tl+qVp2. D, U- log(T'+ U'Vp2D) log(T+ U V/D) where T, U is the fundamental solution of t2 -Du2 = 1, and T', U', of t2 - Dpl2= 1. Since only one form (a, b, c) can be carried over into a particular form (a', b', c') by (1), Dirichlet's42 ratio h(S2D)/h(D) follows at once.43 Similarly, Lipschitz obtained the ratio of the number of improperly primitive classes to the number of properly primitive classes for the same determinant.44 L. Kronecker45 stated that if n denotes a positive odd number >3 and K denotes the modulus of an elliptic function, then the number of different values of K2 which admit of complex multiplication by V -n [i. e., for which sn2(uV/ -n, K) is rationally expressible in terms of sn2 (u, K) and K] is six times46 the number of classes of quadratic forms of determinant -n. These values of K2 are the sole roots of an algebraic equation with integral coefficients, which splits into as many integral factors as there are orders of binary quadratic forms of determinant -n. To each order corresponds one factor whose degree is six times the number of classes belonging to that order. The two following recursion formulas47 and one immediately deducible from them are given. Let n= 3 (mod 4); let F(m) be the number of properly primitive classes of - m plus the number of classes derived from them; 42 Jour. fur Math., 21, 1840, 12. See Dirichlet.20 43 For details, see G. B. Mathews,2l8 Theory of Numbers, Cambridge, 1892, 159-166; also H. J. S. Smith's Report.79 44 For details see G. B. Mathews,218 Theory of Numbers, 1892, 166-169. 45 Monatsber. Akad. Wiss. Berlin, Oct., 1857, 455-460. French trans., Jour. de Math., (2), 3, 1858, 265-270. 46 Cf.. J. S. Smith, Report Brit. Assoc., 35, 1865, top of p. 335; Coll. Math. Papers, I, 305. 47 Cf. L. Kronecker, Jour. fir Math., 57, 1860, 249. 106 HISTORY OF THE THEORY OF NTUMBERS. [CHAP. VI +(n) be the sum of the divisors of n which are > Vn; +(n) be the sum of the other divisors. Then (I) 2F(n) + 4F(n-2 2) + 4F(n- 42) +... =-(n) -(n), (II) 4F (n- 12) +4F(n- 32) +4F(n-52) +... =+-(n) + (n), where, in the left members, n-i2> 0. Using the absolute invariant j instead of K2, H. Weber48 has deduced in detail a similar relation which these two imply.214 C. Hermite49 set u = ( (o) =KI, K being the ordinary modulus in elliptic functions, and found that the algebraic discriminant of the standard modular equation for transformations of prime order n, Inl (8 8(81(to + 6- ) 0, = n, 8 m=o,,...,,-1, is of the form un+l ( -uG8 ) n+ (2/n) 2 (u), where 0 (u) -= a au + a a2u8 u +... + aU8s is a reciprocal polynomial with no multiple roots and 0(u) is relatively prime to u and 1- u8; moreover, v 8 + By means of the condition for equality of two roots50 of the modular equation, he set up a correspondence between these equal roots and the roots of certain quadratic equations of determinant -A and so proved the following theorem.51 Let A'- =(8-3n) (n-28) > 0, " =88 (n- 8;8) >0, O'" - (n- 16s) >0. Then v= -2F(^') + ~2F(Ax") + 6EG (d"'). (Cf. H. J. S. Smith, Report Brit. Assoc., 1865; Coll. Math. Papers, I, 344-5.) Those roots x= (o) of 0(u) =0 are now segregated which correspond to the roots W of a representative system of properly primitive forms of a given negative determinant -A; similarly for a system of improperly primitive forms. If the representative form (A, B, C) of each properly primitive class is chosen with C even, A uneven, then to the roots o of the equations A2 + 2Bw+ C+ = correspond values of u'8 =8 (w) which are the principal roots of a reciprocal equation F (x, A)= 0 with integral 48Elliptische Functionen und Algebraische Zahlen, Braunschweig, 1891, 393-401; Algebra, III, Braunschweig, 1908, 423-426. For the same theory see also Klein-Fricke, Elliptischen Modulfunctionen, Leipzig, 1892, II, 160-184. 49Comptes Rendus, Paris, 48, 1859, 940-948, 1079-1084, 1096-1105; 49, 1859, 16-24, 110-118, 141-144. Oeuvres, II, 1908, 38-82. Reprint, Paris, 1859, Sur la theorie des equations modulaires et la resolution de l'equation du cinquieme degre, 29-68. 50 Cf. C. Hermite, Sur la theorie des equations modulaires, 1859, 4; Comptes Rendus, Paris, 46, 1859, 511; Oeuvres, II, 1908, 8. Cf. also H. J. S. Smith, Report Brit. Assoc., 1865, 330; Coll. Math. Papers, I, 299. For properties of the discriminant of the modular equation, see L. Koenigsberger, Vorlesungen iiber die Theorie der Elliptischen Functionen, Leipzig, 1874, Part II, 154-6. 51 For an equivalent result see Kronecker.124 CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 107 coefficients and of degree the double of the number of those classes. Moreover, F(x, A) can be decomposed into factors of the form r2m2-1 2n-9, 2m-25,...; (x+1)4+ax(x-1)2, if A= 2n, 2 n-4, 2n-16,.... This illustrates the rule that, excepting A=1, 2, the number of properly primitive classes of -A is even if A is 1 or 2 (mod 4). In a theorem analogous to the preceding and concerning improperly primitive classes,, (x, A) =0 is a reciprocal equation with integral coefficients and of degree 2 or 6 times the number of those classes, according as A -1 or 3 (mod 8); and, (x,A) can be decomposed into factors of the form (X2_x+1)3+a(X2-X)2, if A =4n-1, 4n-9, 4n-25,.... For a few small determinants the class-number is exhibited as by the following example. After Jacobi, the modular equations of orders 3 and 5 respectively are (q-)4 =64(1- q2) (l-_2) (3+ q), (q-l)6-256(1-q2) (1-12) [16ql(9-ql)2+9(45-ql) (q-1)2] where q= 1- 2K2, = 1- 2X2. These equations combined with u8 - -x, where /c2 _ - X82, 1 —v8 give, respectively, [x2_x+1][(x2_-+1)3+27(x2_-)2] -s(x, 3).,(x, 11) =0, [ (2 -x+ 1) 3+ 27 (X2 x)2] [(X2 —x + 1)3+ 27.33 (X2 -) 2] -~(x, l)- (x, 19) =0. The common factor of the two left members must be identical with 7F(x, 11). Then the numbers of improperly primitive classes of determinants -3, -11, -19 are one-sixth of the degrees of the expressions in brackets in the left members of the last two equations. P. Joubert's52 modification of this method is given for determinants -15, -23, -31. F. Arndt53 wrote +(x) =II (x-e 2ariP), ~(x) = -II(x-e2bi/P) a b and, in the three cases which Dirichlet had omitted (see Dirichlet,23 (9)), obtained the following: (II) D=P, P=4-++3, (T+UVP)h= -(i)4, where T means - or + according to P is or is not prime; (III) D=2P, P=4iA+l, (T+ U/2P)')^= (x) 4',(-x) 4 (IV) D=2P, P=4,~+3, (T+Ul/2P)h,-=(x)4. (x3)4 J( 52 Cf. Joubert,62 Comptes Rendus, Paris, 50, 1860, 911. 53 Jour. fur Math., 56, 1859, 100. 8 108 HISTORY OF THE THEORY OF NUIMBERS. [CEAP. VI L. Kronecker54 published without demonstration eight class-number recursion formulas derived from singular moduli in the theory of elliptic functions.55 They are algebraically-arithmetically independent of each other; and any other formula of this type derived from an elliptic modular equation49 is a linear combination of Kronecker's eight. He employed the following permanent56 notations. n is any positive integer; in any positive uneven integer; r any positive integer 8k —; s=8k+1>0. G(n) is the number of classes of determinant -n; F(n) is the number of uneven classes. X(n) is the sum of the odd divisors of n; (n) is the sum of all divisors. (n) is the sum of the divisors of n which are > Vn minus the sum of those which are < Vn. '(n) is the sum of the divisors of the form 8k ~1 minus the sum of the divisors of the form 8kS ~3. 4I'(n) is the sum both of the divisors of the form 8/c~ 1 which are > Vn and of the divisors of the form 8k ~3 which are <V'n minus the sum both of the divisors of the form 8k ~1 which are <V n and of the divisors of the form 87 ~ 3 which are >Vn. ) (n) is the number of divisors of n which are of the form 47+ 1 minus the number of those of the form 4k-1. i (n) is the number of divisors of n which are of the form 3 + 1 minus the number of those of the form 37c-1. +'(n) is half the number of solutions of n=x2+64y2; and i' (n) is half the number of solutions of n=x2+3.64y2, in which positive, negative, and zero values of x and y are counted for both equations. (I) F(4n) +2F(4n- 12) +2F(4n-22) +2F(4n- 32) +... =2Z (n) + (,n) + (n), (II) F(2m) +2F(2m-_12) +2F(2nm-22) +2F(2m-32) +... =2@J(m) +~(m), (III) F(2m) )-2(2m-12) +2(2m-22)-2F(2m-32) +... = - (m), (IV) 3G() +6G(m-12) +6G(m-22) +6G(m-32) +... =-(mr) +3@(m) +30<(m) +2 +(m), (V) 2F(m) +4F(m-21) +4iF(im-22) +4F(m- 32) +.. =@(m) + (M) + (m), (VI) 2F(m)-4F(mn - 12) +4F(m-22) -4F(nm-32) +... m -1 = (-1)2 - [(m) - (m)] ++(m), (VII) 2F(r)-4F(r-42) +4F(r-82) -4F(r-122) +... = (- J_ )+(fr-7) () -_,' () ] (VIII) 4 (-1) ] [ ( )-3 ()- ) = ( - )(8"-") [' (s)-~'(s)] + f (s) +4 (s)-40'(s)-8+'(s) 54Jour. fur Math., 57, 1860, 248-255; Jour. de Math., (2), 5, 1860, 289-299. 55 Demonstrated by the same method by H, J. S. Smith, Report Brit. Assoc., 1865, 349-359; Coll. Math. Papers, I, 325-37.100 56 Later in the report of this paper will be noted the historical modification of Kronecker's F and G printed in Roman type. CHAP. VI] BINARY QUADRATIC FORMI CLASS NUMBER. 109 In all recursion formulas (except those of G. IHumbert355) of this chapter, the determinants are ~ 0. In the above 8 formulas, F(0) = 0, G (0) =. The functions +(n), (n), +'(n), +'(n) are removed hereafter from the formulas by replacing italic letters F and G throughout by Roman letters F and G, which agree respectively with the earlier symbols except that F(0) =0, G(0) = - -, and except that classes (1, 0, 1), (2, 1, 2) and classes derived from them are each counted as ~ and - of a class respectively. Later writers have commonly adopted these conventions but have not insisted on printing the symbols in Roman type. The following also result57 from the theory of elliptic functions: F(4n) =F(4n), for all n; F(4n) =2F(n), G F() =F (4n) +G(n), for all n; G(n) =F(n), if n _1 or 2 (mod 4); 3G(n) =[5-(- 1)(n-3)] F(n), if n-3 (mod 4). By means of these relations, Kronecker obtained from the original eight formulas the following58: (IX) F(n) +F(n-2) +F(n-6) +F(n-12) +F(n-20) +... - = (4n+l), (X) E(n) +2E(n-l) +2E(n-4) +2E(n-9) +... =-[2+ (- )"](t), where E(n)=2F(n) -G(n). But a(2~ )zX(n (l)- n=q, 2,..., the plus or minus sign being taken on both sides according as n is even or odd. Hence formula (X) is equivalent to the important formula + 00oo (XI) 12SE(n) q= ~(q), 6,(q) = + q, q=-e,.n= — 0 which implies that the number of representations59 of n as the sum of three squares is 12E(n). (Cf. this History, Vol. II, 265.) By (VI) and (VII), Kronecker calculated F (m) for m uneven from 1 to 10,000. P. Joubert,60 referring to a conjecture of Gauss,61 proved that if n is a fixed prime and A>0 grows through a range of values which are quadratic residues of n, then the number of classes in a genus of the forms of determinant -A has a lower limit for the range. P. Joubert62 considered the principal root o f Po2 + Q +R= 0. If o) furnishes a root 42(o) of the modular equation for transformations of order 2,, /L arbitrary, he found that just two values of 02(0) are furnished as roots by all the forms (P, Q, R) of a given improperly primitive class which have third coefficients a 57 For the means of immediate arithmetical deduction, see Lipschitz 41 and H. J. S. Smith, Report Brit. Assoc., 1862, 514-519; Coll. Math. Papers, I, 246-51. 58 See H. J. S. Smith, Report Brit. Assoc., 1865, 348; Coll. Math. Papers, I, 323. 59 Cf. C. F. Gauss,4 Disq. Arith., Arts 291-2. For a report, see this History, Vol. II, 262; while on pp. 263, 265, 269, are reports on papers by Dirichlet, Kronecker and Hermite giving applications of class-number to sums of three squares. 60 Comptes Rendus, Paris, 50, 1860, 832-837. 61 Disq. Arith.,4 Art. 303. 62 Comptes Rendus, Paris, 50, 1860, 907-912. 110 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI multiple of 16. If (A, B, C) is a form of this kind and of negative determinant -A = (S2 _ 2-+2)/T2, in which S, T are odd, it is equivalent to (2A A, B, C/2p), and these two forms give the same value of 02(0). Consequently, if in the ordinary modular equation we set u2= v2=x, the resulting equation f(x)=0 has a degree which is double the number of representative improperly primitive forms (A, B, C) of negative determinant -A; and f(x) can be decomposed into polynomial factors each of degree the double of the number of the improperly primitive classes of the corresponding determinant - A. For example, if u= 1, the only possible determinant is -T. The modular equation for transformations of order 2 is v4 = 22/ (1 + u4), and becomes x2 + x +2 = 0. Therefore there is a single improperly primitive class of determinant - 7. For somewhat larger values of determinant - (8k-1), Hermite's49 device is used for identifying common factors which belong to the same A and which occur in the left members of f(x) =0 for neighboring values of n=-2'. In the modular equation F(X, K) =0 for transformations of odd prime order n, Joubert wrote X=2x/(l+x2), K=x2, and obtained f(x) =0 in which f(x) is a product of polynomials which have the same characteristic properties as in the former case. If o is such that 02(,) = VK is a root of F (X, K) = 0, then o is the principal root of an equation AoW2 + 2Bo + C = 0, where (A, B, C) is improperly primitive and the negative determinant -A has A equal to one of the numbers n -12, 8n- 32, 8n —52,.... Moreover, C is divisible by 16 and again there are therefore just two values of 2( o) for each improperly primitive class; and the roots 02 (0) lead to forms (A, B, C) which just exhaust the classes of negative determinants -(8n- o2). Hence the aggregate number of improperly primitive classes of the sequence of determinants is read off as in the following example. Let n=3; then =23, 15, F(X, K) X4 -4X3(4K3-3K) +6X2K214+X(3K- 4K) K4, f(x) (4+4x3 + 5x2 +2x + 4) (X6 - 5+ 9X4+13x3+18x2+16x+ 8). Since 15=2.8-1, the first factor in f(x) has already been associated with A =15 by the use of n = 2L = 2. The number of improperly primitive classes of determinant -23 may be read off as half the degree of the second factor and also as the index of its constant term regarded as a power of 2. Joubert63 illustrated his method by many examples. Joubert,64 in the modular equations for transformations of odd order n=paqr7Y (p, q, r different primes), and with the roots vs=s8 (go + 1 6m) added to the usual conditions the restriction that g and g, be relatively prime. In the modular equation f(x, y) =0, he took y = /x. Now f(x, l/y) is of degree 2N=2palq-lr'-l (p+ 1) (+ 1) ( +1) 63 Comptes Rendus, Paris, 50, 1860, 940-4. 64 Ibid., 1040-1045. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 111 and has 2 f+ +/Vn or 2. roots equal to unity, according as n is or is not a square, where r S"y * - (d)), d y d2 ranging over the square divisors of n (n omitted if it is a perfect square), while yyl=n/d2, y< Y, y and Y1 being relatively prime. Excluding the unity roots, he established a correspondence between the roots of f(x, l/x)=0 and the roots of certain quadratic equations, and obtained the following formula when n is or is not a square respectively: F(n) +2F(n- 12 ) +2F(n-2 2 ) +2F(n-32) +... N-* / or N- 2-,+(Vn). where F(D) denotes the number of odd classes of determinant -D which have all their divisors prime to n. If, however, a form is involved which is derived from (1, 0, 1), the right member in each case should be diminished by the number of proper decompositions of n into the sum of two squares. Numerous65 other classnumber relations in the modified F and a similarly modified G are obtained. Tables66 verify the formulas in F. The interdependence of Joubert's and Kronecker's54 classnumber relations has been discussed by H. J. S. Smith.67 H. J. S. Smith68 reproduced the principal parts of the researches of Gauss4 and Dirichlet9, 20, 23 on the class-number of binary quadratic forms. For D>0 and 1 (mod 4), he wrote 7 /T^ / 2\ 2 I Dm M hb(D) (D= )log(TiU r ) _-(~) - log tan mSD llog r(TS+ UN/ --- I.2D' where m is positive, odd, prime to D, and <D. (Cf. Berger,166 (3).) C. Hermite69 began with the factorization H2 (z) 1()_ H(z) () H(z) 02(z) - (z) (z) and expanded each factor after C. G. J. Jacobi,70 setting z= 2Kx/r. In the product of the two expansions, the term independent of x is* ( 1 ), S = E^ W nl. qa ( 2ns+l)aa Ey _(_N) qN 2/4a ~(1) *q/_" Ld..__ qa',+z.~+ F(N) q where in the first sum, a=0, + 1, +2,.... n; while in the second sum, N ranges over all positive numbers =3 (mod 4) which can be represented by (I), and hence by each of the three identically equal expressions (I)2n) (2n+1)(2n+4b+3) -4a2, (II) (2n+1) (4n+4b+4-4a) - (2n+1-2a)2, (III) (2n +1)(4n+4b+4+4a)-(2n+l+2a,)2. *The expansion of the first fraction in (1) is Zqk, k=~(2n+-1)+(2n+-1l)b, b>O. 65 Comptes Rendus, Paris, 50, 1860, 1095-1100. 6 Ibid., 1147-1148. 67 Report Brit. Assoc., 35, 1865, 364; Coll. Math. Papers, I, 343-4. 68 Report Brit. Assoc., 1861, 324-340: Coll. Math. Papers, I, 1894, 163-228. 69 Comptes Rendus, Paris, 53, 1861, 214-228; Jour. de Math., (2), 7, 1862, 25-44; Oeuvres, II, 1908, 109-124. 70Fundamenta Nova Funct. Ellipticarum, 1829, ~~ 40-42; Werke, I, 1881, 159-170. 112 1IISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Thus F(N) denotes the number of ways in which N can be represented by any one of the expressions (I), (II), (III). We represent N by (I), (II), or (III), according as 1a1 <(2n+1 ); a- a ~(2rn+l); a<O, but lai > ~(2n+ 1). Now (I), (II), (III) are respectively the negatives of the determinants of the quadratic forms (2n+1, 2a, 2n+4b+3), (2n+l, 2n+1-2a, 4n+4b+4-4a), (2n+1, 2n+1+2a, 4n+4b+4+4a). Thus we have P(N) forms which are reduced. Moreover, the F(N) forms exhaust the reduced uneven forms of determinant -N. For, those of the first type constitute all uneven reduced forms of determinant -N which have an even middle coefficient. Those of the second and third types constitute all forms (p, q, r) of determinant -N in which p and q are uneven, p>2q, r>2q>0. hence, since (p, q, r) is here never equivalent to (p, -q, r), the number of forms of the three types together is F(N), in the class-number sense.54 A second factorization yields70 (2) (2kK '1() H z (z) 27\ 7r (Z) ~(z) (Z)_____ =,z~% (z) -2 cos 2nz. q7( 1t+ 3 V q- +... + (2 -1) V q('-1'2). For x -0, the first member vanishes and the terms under the summation sign are of the type qN/4. (Ed'- d), where N1-3 (mod 4), d' is any divisor > VN of N and d is any divisor <V'N. In Kronecker's54 symbols, we get, by (1) and (2), OJ (O ):4 (N) q It = 14e* (N) qN14 Or, since ~ (0) = 1 + 2q +2q4+2q9+2ql6+..., (3) F(N) +2F(N-22) + 2F((N- 42) +... -,(). In Kronecker's54 formulas this is (V) + (VI). A third factorization combined with the first yields the following: (4) F(4n- 1) +F(4n- 32) +F(4n- 52) +... = i, (n) - (), where ib(n) denotes the sum of the divisors of n whose conjugates are odd, and * (n) denotes the sum of all the divisors <m/ and of different parity from their conjugates. Similarly, (5) F(N)-2F(N —22) +2F(N-42)... +2(-1)k-(N 4)... = (1)N-32( (-1)1(N-3).1(>(DN) -I(N)), N=3 (mod 4), where '2 (n) denotes the sum of the divisors of n which are < Vn. Hermite's three class-number relations above are all derivable from Kronecker's71 eight. 7i See H. J. S. Smith, Rep. Brit. Assoc., 35, 1865, 364; Coll. Math. Papers, I, 1894, 343. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 113 Since N in (1) is of the form 4n+ 3, (1) implies 0 where c= (1+ )/V2, e4 — 1, and j is the result of replacing q by -q in. Another expression for ~ (<- c) is found by means of the integral of the product quoted at the beginning of this report; comparison of it with (6) gives (7), (8n + 3) q (8n+3) = (- /+ Vq9 + V q25 + 3. This result is implicitly included72 in Kronecker's54 (XI) and can be deduced from it by elementary algebra.73 When the coefficients of equal powers of q are equated in the two members, this formula implies that the number of odd classes of determinant - (8n+ 3) is the number of positive solutions of Sn+ 3 x2y2 +2. L. Kronecker74 referring to his54 earlier memoir, multiplied formulas (I), (II), (V) respectively by q4n, q2, ~qm, added the results, and summed for all values of n and m, and obtained (1) Z(~)p"= al/ = q~+F n -~) F( ) ^W)q, - ~ / K -q2K- ' ( q- +n 2-n) Similarly from formulas (I), (III), (VI), he obtained (2)(n) 7(^" i ^^_ 2nq j-f (2).vF(n)q'n- 2 4 _ n(-q). q-qNow (1) and (2) imply the following three formulas75: K K1 KK kK YF(2m)< = /, = Y (4n+1)q 7-r K r, (8n + 3) q2= k kK and these imply Kronecker's54 (IV). By means of an expansion76 of sin2 am 2Kx/7r in terms of cosines of multiples of x, (1) takes the form (3) 1F(n)q=. k2K | sin2 am, 02(x) cosxdx. q 2r\ f 27r JO 7a "From (3), all the formulas54 (I)-(VIII) can be deduced." Other such relations are indicated by means of theta-functions, although the eight formulas " are algebraically-arithmetically independent." 72Jour. fur Math., 57, 1860, 253. 73 Cf. L. J. Mordell, Messenger Math., 45, 1915, 79. 74 Monatsber. Akad. Wiss. Berlin, 1862, 302-311. French transl., Annales Sc. Ecole Norm. Sup., 3, 1866, 287-294. 75 Cf. C. Hermite,69 Comptes Rendus, Paris, 53, 1861, 226. 76 Cf. C. G. J. Jacobi, Fundamenta Nova, 1829, 110, (1), Werke, I, 1881, 166. 114 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Kronecker stated that he had obtained arithmetical deductions of certain of his class-number relations by following the plan of Jacobi77 who had first found by equating coefficients in two expansions, the number of expressions for n as the sum of four squares and had later translated the analytic method into an arithmetical one.78 The following theorem, which Kronecker deduced from his formula (V), was offered as a suggestion for a means of deducing his class-number relations arithmetically: Let p be any odd prime and let a1z +2b1 z+ c 0O, a2z2+2b2z+c2 O... (mod p) be a succession of congruences corresponding to reduced forms of determinants, -p, - (p- 12), - (p-22),... respectively (with b taken negative in the reduced form if a= c); then the number of roots of the congruences is F(p) +2F(p-12) +2F(p-22) +2F(p-32) +...; that is to say, by formula (V), the number is p+-1 or p according as p is 1_ or 3 (mod 4). I. J. S. Smith79 gave an account of Lipschitz's41 method of obtaining the ratio of h(D.S2) to h(D). C. Hermite80 gave a list of expansions of quotients obtained from theta-functions and showed how the products and quotients of theta-functions lead to class-number relations (cf. Hermite69). This list of doubly periodic functions of the third kind has been extended by C. Biehler,81 P. Appell,s81 Petr,252' 258 Humbert 293 and E. T. Bell.82 Finally, Hermite deduced Kronecker's54 relation (XI). Hermite63 generalized a theorem of Legendre (this History, Vol. I, 115, (5)) into the Lemma: If m =aabfc...kK, where a, b, c,..., 1c are u different primes, then the number of integers which are less than or equal to x and relatively prime to m is (x) =,E x) -E 2:-_E + Eab-:EE +. -.(. _ E @ \abb )E abc) ~+**( abc...1 with the convention >(x) =E(x) if mn=. It follows that (1) 1 (x) = c (m) + 2-1E, -1<e <+. Now F(n) is defined by F(n) =Si=f (i), where f(i) =0 if i is not a divisor of n or if i is a divisor of n but is not prime to m; also F(n) =0, if m and n are not relatively prime. Then, by definition, I n / GF(I)- _f(;)( ) k=l 1 77 Fundamenta Nova, 1829, Art. 66; Werke, I, 1881, 239. 78 Jour fur Math., 12, 1834, 167-172; Werke, VI, 1891, 245-251. 79 Report Brit. Assoc., 1862, ~ 113; Coll. Math. Papers, I, 1904, 246-9. 80 Comptes Rendus, Paris, 55, 1862, 11, 85; Jour. de Math., (2), 9, 1864, 145-159; Oeuvres, II, 1908, 241-254. 81 Thesis, Paris, 1879. s8a Annales de I'Ecole Normale, (3), 1, 1884, 135-164; 2, 1885, 9-36. 82 Messenger Math., 49, 1919, 84. 83 Comptes Rendus, Paris, 55, 1862, 684-692. Oeuvres, II, 1908, 255-263. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 115 Now (cf. Dirichlet,93 (1)), if D =SDo, where Do is a fundamental determinant, and if n is any positive uneven integer relatively prime to D, then for f(i) =(Do/i) and D uneven, for example, the formula (2) F(n) =k ) ( ) k=2 if D<-3, k-= lif D>0, m=2DI, gives the sum of the number of representations of integers from 1 to n which are uneven and relatively prime to D by the representative properly primitive forms of determinant D with the usual restriction84 on x and y in case D> 0. Hermite omits the rather difficult proof that the term containing e in (1) is negligible85 for n very great and concludes from (1) and (2) that, for n very great, F ( i ) - ) - + ( (-i ):+1 2D) "'"'i(-)i +)D k ((-D1)k+l'2). C. F. Gauss86 and G. I. Dirichlet87 had found geometrically the asymptotic mean number of such representations furnished by each form for n large. A comparison yields the class-number (Dirichlet,19 (1)). J. Liouville88 stated that the number89 of solutions of yz+zx+xy= n in positive odd integers with y+z-2 (mod 4), n-3 (mod 4), is F(n). J. Liouville90 obtained an arithmetical deduction of a Kronecker54 recursion formula in the form F(2m-12) +F(2m-32) +F(2m-52) +... =-[2 (m) +p(m)], where m is an arbitrary uneven integer, C (m) represents the sum of the divisors of m, and p(m) is the excess of the number of divisors of m which are -1 (mod 4) over the number of divisors =3 (mod 4). Lemma 1. Let any uneven integer m be subjected to the two types of partitions (1) m = m' + d"", 2m-= m + d2. + 2a~+ld33, where mi, d2, d3, 82, 83 are positive uneven integers; a3>0; while m' is any positive, negative, or zero integer. Then, if f (x) is an even function, Al [d", (m,) - f (2m')- -(2m3') - 2,+) - 2f (2m'+ 4) -. (2m, + t,''- ) ] (2) - If(d2+, d3) d2, + d8). Now take f(x) so that f(0)-=1, f(x) =0 if x -7 0. Then the only partitions of the second type (1) which furnish terms in the right member of (2) are those in which d3=i(d2 + 2). Hence the right member of (2) has for its value the number of solutions of 2m-m =22 s(d2 + ) 2m - m2, = dS, f 82+213(d2 82) 83. 84 G. L. Dirichlet,19 Zahlentheorie, Art. 90, ed. 4, 1894, 225 and 226. 85 Cf. T. Pepin, Annales Sc. de l'Ecole Norm. Sup., (2), 3, 1874, 165; M. Lerch, Acta Math., 29, 1905, 360. 86 Werke, II, 1876, 281 (Gauss 4). 87 Jour. fir Math.,19 19, 1839, 360 and 364. 88Jour. de Math., (2), 7, 1862, 44. 89 Cf. Bell,s70 and Mordell.372 90 Jour. de Math., (2), 7, 1862, 44-48. HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI We set d2 + 82 = 2u, d2 - 2= 4z. Hence u>'2z. Keeping mi fixed, Liouville followed the method of Hermite69 and obtained the result that the number of solutions of 2m-m= u (u+4d) -4z2 is 2(2 m- m() m') -(2- l)-( - (2 -s), m~i mi Si in which g(n) denotes the number of divisors of n, o(n) =1 or 0 according as n is or not a perfect square. Hence Soj (2m-s2) =p(m). Now in the first member of (2), the summation of the first two terms in the bracket is equal to (nm) - (m). Furthermore the expression in (2): f(2mn'+2) +f(2m'+4) +... +f(2m'+8"- ) will have the value 1 for each pair of values m'<O, 2m'+8">0 and the value 0 for all other values m' and 2m '+8". Let A denote the number of pairs of values m'<0, 2m'+8">0 in the partition (1,). We have now proved that (3) 2(2m-m -)- 1E(2m-.m2') - p (m) =- 1 (m)-(m) \-A, Lemma 2. Let any uneven integer M be subjected to the two types of partitions M- 2M'" + D"A", 2M=M - +DA2, where M1, D2, A2, D", A" are positive odd integers, while M' is any integer. Then, if fl (x) is an uneven function, (4) Y +fl M (DI' + 221-') =fl( ) To evaluate A, we identify m and M and specialize f((x) so that fJ(x)=1 if x>0, f1(x)=0 if x=0, f,(x)= —1 if x<0. Since the number of solutions of M= 2M'"+ D"A" with M'>0 is equal to the number with M'<0, the left member of (4) is composed of the following four parts: A=Efl(D" +2MJ'), D"-'<0, "2 >O; -B=:fl(D"-+2M'), M'<0, D"'+2AM'<O; A+B=Sf(-D"+2M'), M'>0, D"+2M'>0; g(m) =f(D"J+r2'), M'=O, D"+2M'">O. Hence (4) implies that 2A+i -(m)= (+ )=ag(2m- 2. Thus (3) becomes91 SF(2m -m( ) =M[2 () +p(M)]. mi This result has been established in detail by Bachmann9l and Meissner.292 From the sanle two lemmas, H. J. S. Smith92 obtains a different form of the right member, for the case m odd. 91 Cf. P. Bachmann, Niedere Zahlentheorie, Leipzig, II, 1910, 423-433. 92 Report Brit. Assoc., 35, 1865, 366; Coll. Math. Papers, I, 1894, 346-350. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. Hermite's discovery69 of the relation between the number of classes of determinant N and the number of certain decompositions of N, also enabled Liouville to announce that formulas exist analogous to those of Kronecker,54 but in which the successive negative determinants are respectively 2s2- n, 3s2 —n, 4s2-n,..., where n is fixed and s has a sequence of values.92" G. L. Dirichlet93 reproduced in a text-book the theory of his memoirs14' 19, 20, 23 of 1838, 1839, 1840. Continuing his former notation, he obtained (Arts. 105-110) new expressions for (P-1)2 N = (8 pjr ( 1 j+ 3r (-L 1)rJ E ( )d SV4, o s X — iron' j = e'i/4, O=e2'1i/P, while s ranges over a complete set of incongruent numbers (mod P) prime to P. The result94 is, for D> 0, D 1 (mod 4), for example, N-2VP=-{_ -( p )-IlogjF()2, F() =n(x-OS) (s/P). Thence in the notation of the Pellian equation, for example, (1) D=P= 1 (mod 8), (T+UVP)(D'=)(t+u2VD)1, 1= [2-(p) (2-K) where K = 1 or 0 according as P is prime or composite, and t, u are positive integers satisfying t2_-Du2=1. From five such relations, Dirichlet points out divisibility properties of h(D); e. g., if D-1 (mod 4), h(D) is odd or even according as P is prime or composite. Incidentally (Art. 91), Dirichlet proved that the number of representations of a number an by a system of primitive forms of determinant D is (2) 7r (D/8) where r= 1 or 2 according as the forms are proper or improper, n is prime to 2D, and 8 ranges over the divisors of n. This formula has been used by Hermite,83 Pepin,l20 Poincare271 to evaluate the class-number. V. Schemmel95 denoted by p an arbitrary positive odd number which has no square divisors. By the use of Gauss sums he set up such identities as the following, when p=4 + 3: 1)) sin 1 p-i /m sin pa sin 2m-r/p (1) -sin m — 1 p 2Vp 1 \p cos 2mr/p - cos a' where a is an arbitrary real number. He took a= r/2 in both members, then A-B-C D D = _>1 P(itanV 2mir A.-B-0+D tD= an 2Vp T \P P 92a Cf. Liouville,107 109 Gierster 145, Stieltjes,15, 162 Hurwitz,167 184 Petr 25, Humbert 293, Chapelon 340. 93 Vorlesungen liber Zahlentheorie, Braunchweig, 1863, 1871, 1879, 1894, Ch. V. 94 Cf. G. L. Dirichlet,23 Jour. fir Math., 21, 1840, 154; Werke, I, 1889, 495; Arndt.53 95 De multitudine formarum secundi gradus disquisitiones, Diss., Breslau, 1863, 19 pp. 118 HISTORY OF 'THETHEORY OF NUMBERS. [CHAP. VI where A, B, C, D are the number of positive quadratic residues which are <p and of the respective forms 4n + 1, 4n + 2, 4n+ 3, 4n+ 4. Whence,96 (2) h(-p) 2 / tan Vp \P P After differentiating both members of (1) with respect to a, he took a=O. The result is23 P-1/m m p-1 /m cot mYr P 2Vp i p whence follows Lebesgue's36 class-number formula (1): (3) p=4n+3, h(-p)= ' cot. p /+, 2vp i(-p)- p Similarly to (3) are obtained (4) p=-4n+1, h(-p) - )2c p=4+3l, h(-2p)- 1 p (m)-P \ sin2mr/ -Vp T \P /cos 4m7r/p 4n, h 1 m) cos 2m7r/p Schemmel, without discussing convergence, decomposed an infinite series by the identity S (K)cosna= lim (-p)jcos ma+cos(p+m)a+.. + (isp+ m) a, K —Pcoo iP where p = 3 (mod 4), and n is positive and relatively prime to p. After transforming the right member, he integrated both members between the limits 0 and ~7r, with Dirichlet's23 formula (82) as the final result: 2 COS cos(b./p+-r/4) (5) h (p) - log 1I log(T+ UVp) cos ( ar/p +7 r/4) Employing the usual cyclotomic notation,53,l(X) =In(x-62a e't), _(X) =nI(z-ee2bvl)), Schemmel found that, for p = 4n + 3 > 0, (6) ipO(1) - '(1) 1 1 - il mcotMr ) - (1) (1) - l-e2a7i/P 1_e2bi/p- - pcot p which by (3) gives a new class-number formula for -p (see H. Holden280). He noted that, for p = 4n + 3>0, cos (^ + c = (n ( ~ a7)r ei7 /; e(b-l-a)r/p 1 r 1, 96 G. L. Dirichlet,23 Jour. fur Math., 21, 1840, 152; Zahlentheorie, Art. 104, ed. 4, 1894, 264. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 119 according as p is composite or prime. Hence by (5), if we set F (x) = log + (x) we have, for p=4 + 3> 0, h(~p)F F(i). (P log(T+ UVp) Similarly for p = 4n + 3 > 0, '(it) 4 (i) 1 p-i(tan2m h ta1nFr h(i) - (i ) Moreover, if p = 4n + 3> 0, h(2p)= log(T+ U/2p) [(i+ ( —2p) = — [Fp () +FWV(-3)] and, if p= 4n + 1, h ___= 4i-[ (2)log(T-+ U 2p) where, in the last four formulas, o = ( + )/V/2. L. Kronecker97 obtained, more simply than had G. L. Dirichlet,98 the fundamental equation (2) of Dirichlet,20 and specialized it in the form (1) TS S(D)(nn')-l-P- > S (ax2+2bxy+cy2)-l-p. a\,,b, c x,y For a particular (a, b, c), the sum co oo S (ax22+2bxy +cy2)-l-P: (x, y) lies between the two values,00' ~P(hy, y) +j hy(x, y)dx, if hy<s<hy+1. Hence 00 00 00 O lim p S p <(x, y) lim p, (x, y)dx, P=O y=l '=1 p=O y/=l hy where h is taken so that ah2+ 2bh + c 0. When we set ax +by=zy, this limit is given by limf0 dz 1 t+u~\/D t D>O p=O ah+b 2-D g 4V t-I/'V u -=^T -, D<0 -.-D Hence, when we exclude99 from the final sum (1) those terms for which the form takes values not prime to P, (1) implies, for p= 0, p( (nn') ----1. 1- log(t +1 )\/(D) - (D) (i )i (, "~ (%) ' VD \.Ip L P - 97 Monatsber. Akad. Wiss. Berlin, 1864, 285-295. 98Jour. fur Math., 21, 1840, 7; Werke, I, 1889, 467. 99Cf. R. Dedekind, Remarks on Gauss' Untersuchungen iiber h6here Arithmetik, Berlin, 1889, 685-686; Gauss' Werke, II, 293-4. 120 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI where p ranges over the distinct prime divisions of P, and t,, u1 are fundamental solutions of t2 DU2-=1. For D>0, this proves that h(D) is finite, since the left member is a definite number. H J.J. S. Smith100 discussed the researches of Kronecker,54' 74 Hermite 49 69, 80 Joubert,62, 64 and Liouville90 in class-number relations. He found proofs of Kronecker's class-number relations64 by means of the complex multiplication of elliptic functions. The details are based on the methods used by Joubert and Hermite. L. Kronecker101 has commended the report for its mastery and insight. For instance, formula (V) of Kronecker is proved by putting x=K2 and l-x= X2 in the ordinary modular equation f8 (K2, X2) =0 for transformations of uneven order m. The right member of the desired formula is found as the order of the infinity of f8(x, 1-x) as x increases without limit. The left member is the aggregate multiplicity of the roots of f8(x, 1-x) =0. R. Lipschitzl02 developed a general theory of asymptotic expansions for numbertheoretic functions and found that, in the special case of the number of properly primitive classes, the asymptotic expression is h(-m) =27 m, s=l, 2, 3,...; m>0. This agrees with C. F. Gauss103 since (2 I+ +... )=-4 1+ ++...) And asymptotically, h(m)= 24 log 2 nm, s=l, 2, 3,...; m>O. The method of Lipschitz is illustrated by C. Hermite.10 J. Liouville065 stated without proof that if a and a' denote respectively the [odd] minimum and second [odd] minimum of the forms of a properly primitive class of determinant - c=-(8n +3) <0, then:a (a'-a) = -7.(- h ). Cl He discussed as examples the cases kc=3, 11, 19, 27. The theorem has been proved arithmetically by Humbert.293 Liouville106 let m be an arbitrary number of the form 8n+ 3, whence the only reduced ambiguous forms of negative determinant - (mn-42) are (d, 0, 8), where d8=m+4or2 and d < V/m-4Uc2. Hence the d's are the values of the minima of the uneven ambiguous classes of determinant - (m-4o2). And hence, if n, [and n2] denotes the number of ambiguous classes of determinant - m whose minima are 1 100 Report Brit. Assoc., 35, 1865, 322-375; Collected Papers, I, 1894, 289-358. 101 Sitzungsber. Akad. Wiss. Berlin, 1875, 234. 102 Sitzungsber. Akad. Berlin, 1865, 174-185. Reproduced by P. Bachmann, Zahlentheorie, Leipzig, II, 1894, 438-459. 103 Werke, II, 1876, 284; Untersuchungen,9 Berlin, 1889, 670. 104 Bull. des Sc. Math.,204 (2), 10, I, 1886, 29; Oeuvres, IV, 220-222. 105 Jour. de Math. (2), 11, 1866, 191-192. 06o Jour. de Math. (2), 11, 1866, 221-224. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 121 (mod 4) [and -3 (mod 4)], and if pi [and P2] denotes the number of uneven ambiguous classes of determinants -(m-4C 2) excluding o-0, whose minima are - 1 (mod 4) [and - 3 (mod 4) ], then in the notation of this History, Vol. II, p. 265 (Liouville33a), -n2-+2(p, - p) =p'(nm) + 2p'(m —4.12) - 2p'(m —4.22) +.... By the theorem there stated, it follows from Hermite69 that F(m) =n1-n2+2(p -p2). Liouville107 stated that he had obtained the following results arithmetically. He generalized Hermite's69 formula (4) both to (1) 2 F(2a+2m-_2)=2a)Sd-D, i>O, i in which i and m are odd; and to (2),i2F(2a+2m-i2) =2am(2a~d-SD)-SD3, i>0 i where a is an integer > 0, d denotes a divisor of m; and D is a divisor of 2am which is of opposite party to its conjugate divisor. By the nature of their second members, these formulas represent what Humbert293 has called the second type of Liouville's formulas. For m = d 12g + 7 or 12g - 11, he gave (3) >F(2mn-3i2) =-1 3+ (3Q)( )d where i=l1, 3, 5.... He stated that if m denotes an odd positive number prime to 5; and a, f are given positive numbers or zero, and m= d8, then (4) >S(8 * 2a5,m - 5i2) = 2a-2 [5+-(- 1) a m 5) i4) where i= 1, 3, 5,..., m = d. A special case of this relation is proved by Chapelon340 as his formula (3) below. If m is a positive integer of the form 24g + 11, then F (m) - 2:(m - 48.s2)- 3d, s>0. Finally, if m= 4g + 3 and t = g-s, then S(8s+3)(8t +3) -=~(- l) (-1/2d2. =o0 The right members here characterize what Humbert has called the first type of Liouville's formulas. G. Humberts08 has deduced formulas of this type, by C. Hermite's method, from elliptic function theory. 107 Comptes Rendus, Paris, 62, 1866, 1350; Jour. de Math., (2), 12, 1867, 98-103. 108 Jour. de Math.,293 (6), 3, 1907, 366-368, 446-447. 122 2HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Liouville109 by replacing n by 3m in Hermite's69 formula (4), decomposed it into two class-number relations 5'F(12m-i_2)= (3m) -,(m), i t0 (mod 3), i=O (mod 2) $F(12m-9i2) = (m), i-O (mod 3), where C (n) is the sum of the divisors of n; and m is odd. Liouvillel10 announced without proof the relation11l ~(-i )iF(4m-n,2) = (a2-4b2), where -=1, 3, 5, 7,...; a is positive and uneven; and a, b range over the integral solutions of m = a2 + 4b2; m odd. Stieltjesl60 and G. Humbertl2 have each given a proof by Hermite's method of equating coefficients in expansions of doubly periodic functions of the third kind. Liouville1l3 stated for m 5 (mod 12) that (1) rF(312%- i3) =4 ( )d where i=1, 5, 7, 11, 13, 17,... is relatively prime to 6; m= d. ForTl4 m odd and relatively prime to 5, F(10m) +2SF(10m - 25t2) = 2(m), where t=1, 2, 3,...;; (m) denotes the sum of the divisors of m. R. Dedekind,15 by the composition of classes, solved completely the Gauss4 problem, obtaining the results of Dirichlet.20 R. Gotting,16 to evaluate Dirichlet'sl4 formula (4) for h(-p), p a prime of the form 4n+ 3, proved that P a\ -1 $(p-)-3) (p-3) (^-) =- P2 +2 j _-r2 pj, a=o jP0 j=P,(p-3 r Z.) 1 i Pa 21 ~where ai = N a+P 12 ' where Uf[j pi + ] P [pi Hence if p=8+ 7, -j = (p2- ); if p=8n+3, j + +2Spj= - (p2 -1). He obtained numerous formulas for computing 2 (a'/p). 109 Jour. de Math., (2), 13, 1868, 1-4. 110 Jour. de Math. (2), 14, 1869, 1-6. i1 Cf. * T. Pepin, Memoire della Pontifica Accad. Nuovi Lincei, 5, 1889, 131-151. 112 Jour. de Math.,293 (6), 3, 1907, 367, Art. 30. 113 Jour. de Math., (2), 14, 1869, 7. 114 Ibid., 260-262. Proved on p. 171 of Chapelon's340 Thesis. 115 Supplement X to G. L. Dirichlet's Zahlentheorie, ed. 3, 1871; ed. 4, 1894, ~~ 150-151. 16 Ueber Klassenzahl quadratischen Formen. Sub-title: Ueber den Werth des Ausdrucks 2(a'/p) wenn p eine Primzahl von der Form 4n +3 und a' jede ganze Zahl zwischen 0 und ~p bedeutet. Prog., Torgau, 1871, 20 pp. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 123 F. Mertens117 denoted by Vl(s, x) the number of positive classes of negative determinants 1, 2, 3,..., x which have reduced forms with middle coefficient ~ s; by x(s, x) the number of these classes which are even. By a study of the coefficients of reduced forms, it is found that the number of uneven classes of negative determinants 1, 2, 3,..., is18 V$/3 F(x)= S [x(s, X)-X(s, X)], 0 where, except for terms of the order of x, Vt7 2/ r Vx/3 _ 7r S q(s,x) 9- z,: X(s, x) slx o 0 0 If we set f(N) - h (-n), we have F(x) =f() (x32) f(f(x/5) +f(x/22 ) +f( ) f(/92) +... F(/32) = f(x/32) + f(x/92)+... F(x/52) = f(x/52) +. and we solve for f(x) by multiplying the respective equations by p (1),, (3), p (5),..., where /juA() is the Moebius function (this History, Vol I, Ch. XIX). Thus f(x)=: p(n)F(x/n2). n=l But \n2 ~ — 1O n2, where Of(x) denotes a function of the order of f(x), or more exactly a function whose quotient by f(x) remains numerically less than a fixed finite value for all sufficiently large values of x. Hence, when terms of the order of x are neglected, r f g(n) n 7rx 1 I 1 I 1 1 f(x)-= 6 - n= 6 33 Then, asymptotically, N 47r 1 Sh (-n) — S N, S =1 + 23 +3+3 +... n= 1 3 33 And therefore the asymptotic median class number is'19 2r./N/ (7S). T. Pepinl20 let >m be the total number of representations of numbers n relatively prime to a given number A, 0 < n _ M, M being an arbitrary positive integer, by a system of properly primitive forms of negative determinant D. He also let Sm be the total number of representations of numbers 2n, n relatively prime to A, 117 Jour. fir Math., 77, 1874, 312-319. Reproduced by P.. Bachmann, Zahlentheorie, Leipzig, II, 1894, 459. 118 Cf. C. F. Gauss, Disq. Arith., Art. 171. 119 Cf. C. F. Gauss, Disq. Arith., Art. 302; Werke, II, 1876, 284. Cf. R. Lipschitz,'02 Sitzungsber. Akad., Berlin, 1865, 174-185. 120 Annales Sc. de 1'Ecole Norm. Sup., (2), 3, 1874, 165-208. 9 124 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 0 ~ n~ < M, by a system of improperly primitive forms of determinant D. In every representation, let x=axi+y, y=fi3y+S, y<a, 8<,, a, /, y, 8 each 0; and in each of the two cases above, let K, K' be respectively the number of pairs of values y, 8 possible for given a, /f. ThenT21 Kh(D) 7M +ME KXh(D)7rM + = ^M ap ap where the limits of Me and My for 21= oc are finite. A comparison of K and K' for a=f8=A=2 gives Dirichlet's20 ratio h/h'. The corresponding result is obtained for the other orders and for the positive determinant. Pepin avoids the convergence difficulty of Hermite83 and obtains Dirichlet's23 classic closed expression (5) for h (D), D<0, by extending a theorem of Dirichlet93 (2), to give:Mn?=K E (Do/i), n i in which K is the automorph factor 2, 4 or 6; Do is a fundamental determinant, D= DOS2; i ranges over all divisors of n, while n ranges over all odd numbers < M; and (Do/i) is the Jacobi-Legendre symbol. Pepin translating certain results of A. Cauchyl22 on the location of quadratic residues, found in Dirichlet's23 notation (1) (h(-n)=)rb _ b2) a2, where -n=-(4/ + 3) is a fundamental negative determinant. This latter classnumber formula, called Cauchy's, has been simply deduced by M. Lerch, Acta Math., 29, 1905, 381. Other results of Cauchy123 give, in terms of Bernoullian numbers, h(-n)- 2B(,n+)/4 if n=81Z+; — 6B(n,+)/4 if n=8Z+3, modulo n a prime. And without proof Pepin states, for n>0, that (-)= [2- ()] [21+l][4+1]-2 q[Vmin], 1= - L. Kronecker124 obtained from his54 eight classic relations new ones, as, for example, by combining (IV), (V), (VI), the following: S(-l)^F(n-4h2) =i-_(-_)'(n-3)jD(n) +.k(,n) n=3 (mod 4), h>0 h By means of125 (1) 4:F (4n + 2) q+=2(q) 3 (q), 0 121 C. F. Gauss, Werke, II, 1876, 280; Untersuchungen iiber hohere Arithmetik, Berlin, 1889, 666. 122 Mem. Institut de France,29 17, 1840, 697; Oeuvres, (1), III, 388. 123 Mem. Institut de France, 17, 1840, 445 (Cauchy28); Oeuvres, (1), III, 172. 124 Monatsber. Akad. Wiss. Berlin, 1875, 223-236. 125 Cf. Monatsber. Akad. Wiss. Berlin,74 1862, 309. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 125 he obtained formulas for,G(s721), 2(Fs- 1), s- I (mod 8). He obtained two analogues156 of (1), and stated that, in his54 classic relations, I(IV) -~ (V) +~3 (VI) -j(VIII) is, when m is the square of a prime, equivalent to Hermite's49 first class-number relation. R. Dedekind126 supplied the details of Gauss'9 fragmentary deduction of formulas for h (D) and h(- D). He also127 deduced and complemented Gauss'9 set of theorems which state, in terms of the class-number of the determinant -p, the distribution of quadratic residues and non-residues of p in octants and 12th intervals of p, where p is an odd prime. Dedekind,l2 a in a study of ideals, obtained results which he translated373 immediately into the solution of the Gauss Problem.4 Dedekind128 extended the notion of equivalence in modular function theory by removing the condition129 that p and y be even in the unitary substitution (e ) ). Each point w in the upper half of the complex plane is equivalent to just one point wo, called a reduced point, in a fundamental triangle defined as lying above the circle x2+y2= and between the lines x= ~+ and including only the right half of the boundary (cf. Smith95 of Ch. I). The function, called the valence of o, ~(i~~) - ^ A^~ 4(7lCp)3(kPT 2) (1) v — va (= ) = — 7 (- k2 (I (ik) P k= K where p is an imaginary cube root of unity, is invariant130 under the general unitary substitution. Dedekind's v is -4/27 times C. Hermite's31 a. Let C~+DO) C D Vn= val(A )- A B where A, B, C, D are integers without common divisor. Then vn ranges exactly over the values vat (C + do ) v a where a, c, d are integers > 0 and ad= n; moreover, if e is the g.c.d. of a and d, then c ranges over those of the numbers 0, 1, 2,..., a which are relatively prime to e. Hence the number of distinct values of v, is a (2) v=- ) -l- ), where p ranges over the distinct prime divisors of n. 126 Remark on Disq. Arith., in Gauss's Werke, II, 1876, 293-296; Untersuchungen fiber Hohere Arithmetik, 1889, 686-688. 127 Gauss's Werke, II, 1876, 301-303; Untersuchungen, 1889, 693-695. 127a Uber die Anzahl der Ideal-classen in der verschiedenen Ordnungen eines endlichen Korpers. Festschrift zur Saecularfeier des Geburstages von Carl Frederich Gauss, Braunschweig, 1877, 55 pp. 128 Jour. fur Math., 83, 1877, 265-292. 129 Cf. H. J. S. Smith,100 Rep. Brit. Assoc., 35, 1865, 330; Coll. Math. Papers, I, 299. 130 Cf. C. F. Gauss, Werke, III, 1876, 386. "31 Oeuvres, II, 1908, 58 (Hermite 49). 126 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Dedekind discussed the equations whose roots are the v values of v,. H. J. S. Smith'32 called the totality of those indefinite forms which are equivalent with respect to his normal substitution (Smith95 of Ch. I) a subaltern class. He found that if a denotes 2 or 1, according as U is even or uneven in T2 -NU2 1, the circles of each properly primitive class of determinant N are divided into 3o subaltern classes which in sets of u satisfy the respective conditions (A) a=c 1 (mod 2); (B) a-O, c-1 (mod 2); (C) al, cO- (mod 2). Since the circle [a, b, c] corresponds to both (a, b, c) and (-a, -b, -c), the number of subaltern classes of properly primitive circles of determinant N is H=-2 h (N). There is a similar relation for the improperly primitive circles. Now = x + iy, representing a point in the fundamental region 2, is inserted in P8(0) + Y, + +' 8 (') 1 -X _- y, where s8(o), J8(w) are Hermite's49 symbols in elliptic function theory. Then if the circle [a, b, c] satisfy (A), for example, the arcs within S of all and only circles (completely) equivalent to [a, b, c] are transformed by the modular equation F(k2, 2) =0 of order N into a certain algebraic curve, an interlaced lemmiscatic spiral. Hence all the circles of determinant N that satisfy (A) go over into a modular curve consisting of ~H distinct algebraic branches. This is called by F. Klein the Smith-curve.l33 The number of improperly primitive subaltern classes of determinant N (not a square) is just the number of branches of a modular curve which is derived as the preceding from circles of determinant N, in which a- c 0 (rod 2). F. Klein134 called Dedekind'sl28 v the absolute invariant J and, instead of v,, he wrote J'. The equation, II(J-J') =0 is called the transformation equation of order n. He gave an account of its Galois group, fundamental polygon, and Riemann surface. Simplest forms of Galois resolvents are found for n=2, 3, 4, 5. For example, the simplest resolvent for n=5 is the icosahedron equation. Define v (w) as a modular function if it is invariant under a subgroup of the group of unitary substitutions (a I). Then wo and 02 are relatively equivalent if (o,) = s(oj2). A subgroup (a I) is said to be of grade (stufe) q if (a (a ) (mod q), (^ c d where a, b, c, d are constants. Klein ascribed the grade q to any modular function which is invariant under only (that is, belongs to) such a subgroup. The subgroup (a,B _ (1 ~0 (mod q) is called the principal subgroup; and it is found that the icosahedron irrationality belongs to this subgroup if q= 5. This result for the case of n= 5 is extended to all 132 Atti della R. Accad. Lincei, fis. math. nat. (3), 1, 1877, 134-149; Coll. Math. Papers, II, 1894. 224-239; Abstract, Transunti, (3), 1, 68-69. 33 Elliptische Modulfunctionen,2'7 II, 1892, 167 and 205. 134 Math. Annalen, 14, 1879, 111-162. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 127 odd primes n. A modular function which belongs to the principal subgroup is called a principal modular function. If n is an odd prime, the simplest Galois resolvent is of order ~n(n2-1) and its Riemann surface is equivalent to ~n(n2-1) triangles in the modular division of the plane. These triangles are chosen so as to form a polygon; and the surface of the resolvent is formed from the polygon by joining the points in the boundary which are relatively equivalent. The genus of the surface is p -(n-3) (n-5) (n+2). Klein hereafter ascribes the p of the surface to - itself. Hence if a principal modular function r has q=3 or 5 then p=O; but if q-7, then p=3. It follows that if q is an odd prime, J is a rational function of r if and only if q=3 or 5. It is found similarly that if q=2 or 4, J is a rational function of -q. The modular equation of prime order n 7= 5 and of grade 5 is written as (1) n[I(W)-v(0)]=0, where (o) is the icosahedron function, and the n+1 relatively non-equivalent representatives '/ are displayed in detail. J. Gierster'35 wrote a set of eight class-number relations which he stated he had found from the icosahedron equation (Klein,134 (1)) by the method of L. Kronecker136 and Smith.100 For example, 3fH(4n-O') =-(n), n- ~1 (mod 5), where, as always hereafter, H(m) denotes the number of even classes of determinant -m with the usual conventions54; c1 ranges over positive quadratic residues of 5 which are ~ V4n. A combination of these eight relations gives (A) H (4n-k2) = (n) + (n), which may be expressed in terms of Kronecker's54 original eight: (n) - II(m) or I (n (m) -- I (m) +IV(m)- -V(m), according as u is odd or even in n= 2'1m, where m is odd. T. Pepin137 completed the solution of Gauss'4 problem. He accomplished this by finding the number of properly primitive classes of determinant S2 D which when compounded with (S, O, -D-S) reproduce that class. Similarly he found the ratio between the number of properly and improperly primitive classes of the same determinant. F. Klein138 emphasized the importance of the study of the modular functions (cf. Klein134) which are invariants of subgroups of finite index (i. e., subgroups whose substitutions are in (1, Jk) correspondence with those of the modular group) and in particular those in which the subgroups are at once (a) congruence sub 135 Gottingen Nach., 1879, 277-81; Math. Annalen, 17, 1880, 71-3. 136 Monatsber. Akad. Wiss. Berlin, 1875, 235. 137 Atti Accad. Pont. Nuovi Lincei, 33, 1879-80, 356-370. 138 Math. Annalen, 17, 1880, 62-70. 128 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI groups, (b) invariant subgroups, and (c) of genus zero. In the last case, a (1, 1) correspondence can be set up between the points of the fundamental polygon of the sub-group in the w plane and the points of the complex plane by means of the equation J=f(-) of genus zero where -7(o) is called a haupt modul. But if the genus p is >0, v() must be replaced by a system of modular functions Mi(w), M2(W),.... Klein and after him A. Hurwitz and J. Giester always chose Mi(ow) so that MT f aWHOS\, ), _ = a, Mi[ — w+9)aJ-f3y=l, for all values of i, is a linear combination of M, (o), M1 ((o),.... The representatives o/are (Aw +B)/D, with AD=n, 0 ~ B<D, B having no factor common to A. and D. The analogue of the vanishing of IIn [(() - 7 ()] in the modular equation'34 for the case p=O, is for the case p>0 the coincidence of the values of M,(w), M2(w),... with those of M (w/), M2(/'),... respectively. This analogue of the modular equation is called the modular correspondence and it is said to be grade q if the M's are of grade q. J. Gierster139 stated that all of F. Kronecker's54 eight class-number relations are obtainable as formulas of grades 2, 4, 8, 16. From F. Klein's'40 correspondence of order n and grade q>2, Gierster obtained r=-q(q2-1) correspondences by means of the unitary substitutions. He also considered the case where A, B, D have a common factor, i. e., the reducible correspondence. The number of coincidences of a reducible correspondence at points co in the fundamental polygon'34 for q can be determined arithmetically in terms of class-number and algebraically in terms of the divisors of n. Excluding the coincidences which occur at the vertices, in the real axis, of the fundamental polygon, he gave briefly the chief material for the arithmetical determination. This he145 made complete later. If a given congruence subgroup G is not invariant, Gierster indicated a method of finding the number of coincidences of a correspondence for G in terms of the number of coincidences of the r reducible correspondences for the largest invariant subgroup under G and hence in terms of a class-number aggregate (cf. Gierster148 for details). He here stated (but later'14 proved) a full set of class-number relations of grade 7 (failing to evaluate just one arithmetical function (n) which occurs in several of the relations). These relations for the case when n is relatively prime to 7 were derived in detail later by Gierster148 and A. Hurwitz142 by different methods, Gierster employing modular functions which belong to other than invariant congruence subgroups. A. Hurwitz143 denoted by D any positive or negative integer which has no square factor other than 1, and wrote F(s,D)=[l(-l)s(D2-1) 1] -(D)1 if D=1 (mod 4), F(s, D) =_ (1-)- in all other cases, fn ns8 139 Sitzungsber. Munchener Akad., 1880, 147-63; Math. Annalen, 17, 1880, 74-82. 140 Math. Annalen, 17, 1880, 68 (Klein 138). 141 Ibid., 22, 1883, 190-210 (Giester 148). 142Ibid., 25, 1885, 183-196 (Hurwitz 84). 1Z3 Zeitschrift Math. Phys., 27, 1882, 86-101. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 129 where the summation extends over all integers n>O prime to 2D. (Cf. Dirichlet,19 (1).) He proved the following four theorems: (I) The functions F(s, D) are everywhere one-valued functions of the complex variable s. (II) Every function F(s, D), except F(s, 1), has a finite value for every finite value of s. (III) For every finite value of s, the function F(s, 1) has a finite value except when s= 1. Then F(s, 1) becomes infinite in such a way that lim [(s-1)F(s, 1)] =1. s —l (IV) If D>O, F(1-s, D)=(2 ) r(s) VKDCoss7 'F(S,D) 7(KD)S - F(s, D); r 2 ~ if D<O, F(1-s, D)=( _7) Fs (s) V —KDsin s7r.F(s, D) F(1-s, D) -, 7r __ ~ ~~=(-,..D) l (s ))- )(s, D) where - = 1 if D- 1 (mod 4), K= 4 in all other cases. These four results are extended to D= Y.S2 by the use of Dirichlet's identity D\ I /ID I n( n ) s ( n) An" WS [ (r )8 ] where n' ranges over all positive integers prime to 2D', and r ranges over all prime numbers which are divisors of D' but not of D (cf. Dirichlet, Zahlentheorie, ~ 100). The memoir ends with an ingenious proof of the three following theorems: If D>0 and D# 1, F(s, D) =0, for s=0 and for all negative even integral values of s. If D<O, F(s, D) =0 for all negative odd integral values of s. F(s, D) *r(2) * (<)"(D>O), F(s,D)~( ) *( ) (iD<0) are not altered in value when s is replaced by 1- s. L. Kronecker144 proved six of his54 eight classic relations by means of a formula for the class-number of bilinear forms and a correspondence between classes of bilinear forms and classes of quadratic forms (Kronecker14 of Ch. XVII). Two quadratic forms are completely equivalent if and only if one is transformed into the other by a unitary substitution congruent to the identity (mod 2). (For 144Abhand. Akad. Wiss. Berlin, 1883, II, No. 2; Werke, II, 1897, 425-490. 130 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI more details, see Kroneckerll3 of Ch. I.) Whence 12G(n) and 12F(n) are the number of classes and of odd classes respectively of determinant -n under this new definition of equivalence. Two bilinear forms are likewise completely equivalent if they are transformed into each other by cogredient substitutions of the above kind. Then the number of representative bilinear forms Axly,+Bxy2- Cx2y1 +Dx2y2 having a determinant A-=AD +BC is 12(G(n)-F(n)) or 12G(n), according as B+ is odd or even where n= -A + (B + C)2 is the determinant of the quadratic form (A, ~(B-C), D). But since G(4n) -F(4n) =G(n), the number of classes of bilinear forms of determinant A is 12a[G(4A-h2) -F(4A- 2) ], _ 2VA<7<2VA. h And there are 12FF(A-h2) classes of those bilinear forms of determinant A, for which at least one of the outer coefficients A and D is odd and the sum of the middle coefficients B and C is even. The class-number of bilinear forms is now obtained in terms of 1 (A), (A) and X(A). This gives immediately such class-number relations as S:[G(4A-2) -F(4A-h2)] =@(A) +- (A); h and so (I)-(VI) of Kronecker.54 J. Gierster145 gave a serviceable introductory account of the modular equation f(J', J) -II(J-J') = 0 and of the congruencial modular equation, and also of the congruencial modular correspondence. He determined (p. 11) the location and order of the branch-points of the Riemann surface of the transformed congruencial modular function o(w') as a function of tu(w), for the case q a prime, n prime to q, and,A () belonging to the unitary sub-group, (1 ) (a) ) (mod q). From the condition that o furnish a root of the reducible modular equation34 f (J', J) = 0, namely, that integers a, b, c, d exist such that (2) ao b ad-bc=, he established (p. 17) a correspondence between the roots of f(J', J) =0 and the roots of certain quadratic equations Po+Qo + R- 0 of all discriminants - A = (d+a)2-4n<O. Whence the number of zeros of f(J', J) in the fundamental triangle is IH (4n —K2), K=0, -+1, + 2..., K2<4n. To study the infinities of f(J, J') in the fundamental triangle, Gierster (after Dedekind128) took o'= (Aw +B)/D, noted the initial terms in the expansion of J and J' in powers of q = e"r, and found that (J-J') D/ = const. Jsl in the neighborhood of o,=ioo; in which g is the greater of A and D, and T is the 145 Math. Annalen, 21, 1883, 1-50. Cf. Gierster 39; Klein-Fricke, Elliptische Modulfunctionen, II, 160-235. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 131 g.c.d. of A and D. Whence, taking into account the number of values of B, he arrived at the class-number relation:H(4n-K2) =(n) +'(n), K-c=, ~+, +2,.... The result also follows from the Chasles correspondence principle.146 The irreducible correspondence'39 is now studied (p. 29) between P/i(o) and /Az(O'), where the i(o) are a system of functions invariant only of the subgroup of unitary substitutions (1), and w' ranges over a complete set of relatively nonequivalent representatives -+~ ad-bc-n, Co + d' where n is prime to q, and a, b, c, d have fixed residues (mod q). Now o in the fundamental polygonT34 furnishes a finite coincidence if and only if there exist integers a, b, c, d satisfying (2). Hence the condition is that o be the vanishing point for some form Po2 + QW + R, for which (3) + ~P ---c, ~Q-+ ~(d-a), ~R_==b (mod q). For an arbitrary reduced form Po2 + Q o + RO, let g be the number of equivalent forms P,,2 + Qyo + R which have roots in the fundamental polygon and which satisfy both (3) and (4) (PvP QV, R.)(a )=(Po, Qo, RO). In the particular case, b - c 0, d = a= 'n, we have a + d = 2 /n, 0 = Po Qo- Ro =A=4n-(a+d)2 (mod q); and (3) and (4) impose no condition on a, /3, y,,. Hence (Klein134), g=iq(q2-1) and the number of finite coincidences is lq(q- ) -q n, where I ranges over the positive and negative integers for which 4n —l2 is positive and divisible by q2, while H'(m) is the number of classes of forms of discriminant - m which have no divisor which is a divisor of n. The number of finite coincidences of the reducible correspondence of order n is therefore z=,q(q2-l)H (4n-K) where K = KVn ranges over the positive integers 2V/n which are ~ 2V/n (mod q). Gierster now finds for the reducible correspondence the number of infinite coincidences in the fundamental polygon. For the above particular case, this is O =Zoo + (q2 - 1) UVn where U4 denotes the sum of the divisors of n which are < V and ~i (mod q), provided that, if Vn is an integer - ~i (mod q) then -V/n is to be added to the sum. He evaluated a in many further cases. 146 M. Chasles, Comptes Rendus Paris, 58, 1864, 1775. A. Cayley, On the Correspondence of Two Points on a Curve, Proc. London Math. Soc., 1, 1865-6, Pt. VII; Coll. Math. Papers, VI, 9-13. 132 HISTORY OF THE THEORY OF NUMBERS. [CHAtP. VI For q's such that134 p=O, the a's are evaluated also by the principle of Chasles.146 And so for q=3 and 5, twelve exhaustive class-number relations are written such as (for our particular case above): =5, ( —)1 60H (42?= (n)-12 UVn. J. Gierster147 tabulated congruence sub-groups of prime grade q of the modular group and calculated their genus (Klein134) for q ~ 13. Gierster148 continued his145 investigation but now replaced his former invariant subgroup of grade q by any one not invariant. There the total number of coincidences in the correspondences was expressed as a sum a of class-numbers. Here the analogues of the o's are found to be mere linear combinations of the former a's. Employing congruence groups of grade 7, 11, 13 and genus134' 138 zero, he deduced class-number relations149 including for example 4:(H(4n-K<2) =- (n), q-7, s=4V-n, (n/7)=-1. A. Berger150 employed an odd prime p, integers m, n and put Un= ( 1) [] =4[ Vn]-2[ [Vn] + 1, O C n<p2, S-= Um+4(- O n<4p, k= where [x] denotes the largest integer _ x. Various expressions for Sm are found. For example, if p 1 (mod 4), k`Im/4 / C\ (1) S-~-|-2 X -, k>(m-p)/4 P where E= +1 if m-0 (mod 4), e= -1 if m 1, 2, or 3 (mod 4). Write k<rp/8 '1C) Lr,= k>(r-l)p/88 P Let KZ be the number of properly primitive classes of determinant -p, and K2 that of determinant -2p. A study of Lr and Dirichlet's23 formula (5) give KZ=2(LI+L2), K2=2(L1-L4), if p 1 (mod 4); L-=L8=j(K1+K2), L2=L4=L5=L7=- (K-KI2), if pl- (mod 8). Whence, for pl 1 (mod 8), he found by (1) such relations as So=1+ - K, Sp= - 1+ -K, Sp= -1- K, S3P=-1-K1, S(P-I)/2 =1+K- +K2, S(3P-1)/2= —1-K1. Similar relations are obtained for p- 3, 5, 7 (mod 8). 147 Math. Annalen, 22, 1883, 177-189. 148 Ibid., 190-210. 149 Notations of Gierster145 (2), or more fully in Math. Ann., 22, 1883, 43-50. 150 Nova Acta Reg. Soc. Sc. Upsaliensis, (3), 11, 1883, No. 7, 22 pp. For some details of the proof of (1), see Fortschritte Math., 14, 1882, 143, where the denotation of (1) is incorrectly given. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 133 Berger wrote Q (x) for the largest square ~ x and deduced eight theorems like the following: Among the p squares (2) Q(0), Q(4p), Q(8p),..., Q4(p-1)p), there are ~(p+1+.K1), ~(p+l), -(p+1+KI), or ~(p+l-2K1) even numbers, according as p 1, 3, 5, or 7 (mod 8). Since K] and K2 are positive, the squares (2) include at least i(p+5), -1(p+l), i(p+3), or i(p-l) even numbers in the respective cases. C. Hermite'51 communicated to Stieltjes and Kronecker the fact that if F(D) denotes the number of uneven classes of determinant -D, then (cf. Hermite,164 (2)) F(3) +F(7) +... + (4n-1) -E(2j ) E( 3 23) 2/ — 2- 47/ i O2-271^. ++ 2SE (nv -4- i. + 2E...2 in which n-v2- 2cv 2v+2k+1. Hermite152 stated Oct. 24, 1883, that if F(N) denotes the number of properly primitive [he meant uneven] classes of determinant -N and q (n) =(- 1 1) ( /2, where d ranges over all divisors < /n of n, then 1(3) +F(11) +F(19) +... +F((n) =-(3) +(11) +... ++(n) + 2a^(lc)E(~~ n- k) +2a()E(\(iVg/-z+i); l=3, 11, 19,..., n; =7, 15, 23,..., n-4. T. J. Stieltjes153 observed that this result is equivalent to F(n) =I,(n) +2 (n-4. 12) +2i (n- 4.22) +2 (n - 4.32) +...; and this is equivalent to an earlier result of J. Liouville, Jour. de Math., (2), 7, 1862, 43-44. [For, by definition, Liouville's p'(n) is Hermite's t(n); see this History, Vol. II, Ch. VII, 265, note 33a.] Stieltjes154 let F(n) denote generally the number of classes of determinant -n with positive outer coefficients, but in case n= 8k + 3 with even forms excluded. Then he found, when n- 5 (mod 8), that IF(n) is the number of solutions of n=x2+2y2 + 2z2, x, y, z each >0 and uneven. Consequently setting (P (n) = (2/d,) d, dd- = n, he found that F(n) +2F(n-8 12)2 +F2(n-8.22) +... = (n), n=3 or 4 (mod 8); F(n-2 12) +F(n-2.32) 4+ F(n-2.52) +...= - (n), n-5 or 7 (mod 8). On Nov. 15, 1883, Stieltjes155 observed that the former of the last two theorems is a corollary to Gauss, Disq. Arith., Art. 292. For i=l1, 2, 3, 5 or 6, he found that N + 7:F(8 + i) =- N n=1 48 151 Aug., 1883, Correspondance d'Hermite et Stieltjes, Paris, I, 1905, 26. 152 Correspondance d'Hermite et Stieltjes, Paris, I, 1905, 43. 13 Ibid., 45; Oct. 28, 1883. 154 Correspondance d'Hermite et Stieltjes, Paris, 1905, I, 50-52, Nov. 12, 1883. 155 Ibid., 52-54. 134 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI asymptotically (cf. Gauss,4 Disq. Arith., Art. 302, Mertens,/17 Gegenbauer,199 Lipschitzl02). Stieltjes,l56 by the use of the two Kronecker124 formulas, 4:F(4n+ 1) q+=02(q) 6(q), 8:F(8n+3) q2n+-=0(q), o 0 obtained the following three results: Let b(n) =S(2/d')d, dd'=n;,(n) =:(-2/d), whence 2T (n) is the total number of representations of n by x2+ 2y2; then n-1 (mod 8), S(n-8r2) =-4(n)+-+(n) - n= 3, 5 (mod 8), F (n-8r2) = - (n) n 3, 5, 7 (mod 8), SF(n-2s2) =i (n) + '(n), (s= 1, 3, 5,...). Stieltjes157 stated that he had deduced Liouville'sl10 class-number relation of 1869 and other similar formulas both by arithmetical methods and by the theory of elliptic functions. For example, for NV>0, 2~:(-1)(8-1) sF(4N- 2s2) = ( -1)IN(N-1):S(x2_y2), s==l, 3, 5, summed for all integral solutions of 2+- 2y2=N. This he'58 later proved in detail. For N>0, (-1 )i(8s-)sF(16N-3s2) =- (X2 - 3y2), s=1, 3, 5,..., summed for all integral solutions of different parity of 2 + 3y2 =N. The method of verifying this formula was indicated159 later. Stieltjes160 obtained from classic expansions the expansion (1) 0(q)O(q)03(q)=16 (x2-y2)q2y, x=, 3, 5,,... y=O, ~2, +4,.... But (2) 0(q)02(q)0(q) = 2(q/4+39/4+525/4-..; and (cf. Hermite,69 (7)) 0 (3) 6(2()= 8S (8n + 3)) qt8+3) A comparison of (1), (2), (3) gives at once a Liouville10 class-number relation. Stieltjes added three new relations of the same type; e. g., for N= 8k + 1, -2: ( _ 1 ) (8-l)>+(2-sl)s(2N-s2) = (- 1) (2- 8y2), summed for all integral solutions of x2+ 8y2 = N in which x> 0 and uneven. Stieltjes'61 stated, for the Kronecker54 symbol F(n), that (1) F(np2) )= [p +pp-I +... _ (n-n) (-+p —2 +.....)](n). 56 Correspondance d'Hermite et Stieltjes, Paris, I, 1905, 54, Nov. 24, 1883. 157 Comptes Rendus, Paris, 97, 1883, 1358-1359; Oeuvres, I, 1914, 324-5. 158 Correspondance d'Hermite et Stieltjes, Paris, I, 1905, 63, Nov. 27, 1883. 5 Ibid., 69-70, Dec. 8, 1883. 160 Comptes Rendus, Paris, 97, 1883, 1415-1418; Oeuvres, I, 1914, 326-8. 161 Correspondance d'Hermite et Stieltjes, Paris, I, 1905, 81, 85-87, letter to Hermite, Jan. 6, 1884. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 135 He gave162 a proof depending on the fact that F(n) =pSh(n/d), where d ranges over the odd square divisors of n; p=1 or 1, according as n is or is not an uneven square; h(m) denotes the number of properly primitive classes of determinant -m. Stieltjes162 put 7(n) =5,(-)(d-18d (ddl =n); X(n) =Sx where x ranges over the solutions of n=x2-2y2>0, x>0, Iyl< x; and stated that, when n is odd, 2,(_-)rF(n-2r2) = (-_)( -1)2x(n) r-= 10, 2,...; 2:2F(n- 2r2) =2+(n)-X(n), r=0, -1,... These and two similar formulas he was unable to deduce by equating coefficients of powers of q in expansions. This was later done for formulas which include these as special cases by Petr,258 Humbert,293 and Mordell.352 C. Hermite163 imparted to Stieltjes in advance the outline of the deduction of Hermite's164 formula (1). Hermite,164 by the same study of the conditions on the coefficients of reduced forms as he employed69 in 1861, found that (q) = 24 [ (N) + 2f (N) qN14-16c, where (N) denotes the number of ambiguous, and f (N) the number of unambiguous, even classes of determinant -N; while E=1 or 0, according as N is or is not the treble of a square. For the case N 3 (mod 8) a comparison of this with his earlier result69 62(q)=-8:F(N) qN/4, where F(N) is the number of uneven classes of -N, gives at once the ratio between the number of classes of the two primitive orders (cf. Gauss,4 Disq. Arith., Art. 256, VI). Kronecker's124 formula (1) implies that (1) 6O(g)(q) =4(1gqn)F(4n+2)q"+n=4:[F(2) +F(6) +... F(4n+2)]q-. 1 —q o o n0 But obviously 20(q) [2S q2(2n+l)2]2 =f(8c+2) q2-, O o where f (n) denotes the number of solutions of X2 + y2 = n. Moreover, 03(q) =1+ 2Sql. Therefore, in the identity 0(q)6 (q) _ 6(q) +2 (nqn)62(q) 1-q - 1-q 1-q ' the first term of the right member is Sf(8c+ 2)q'+1, summed for n0=, 1, 2,...; 162 Correspondance d'Hermite et Stieltjes, Paris, I, 1905, 82-85, Jan. 15, 1884. 163 Ibid., I. 88-89, Feb. 28, 1884. 164 Bull. de 1'Acad. des Sc. St. Petersburg, 29, 1884, 325-352; Acta Math., 5, 1884-5, 297-330; Oeuvres, IV, 1917, 138-168. 136 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI c=O 0, 1, 2,..., [n]; the second term, by a lemma on the Legendre greatest-integer symbol, is 2Sf(8c+2) [n - 2c]. qq4, summed for n=0, 1, 2,..., c=O, 1, 2,..., [i(n-1)]. Hence a comparison with (1) gives 4[F(2) +F(6) +... +F(4n+2)] = S f(8c1+2)+2Sf(8c+2) [Vin-2c]. cl=0 By the use of Jacobi's expansion formula: 8O (q) = 4 /q q2-4V q + 4 qs 1 +ql _.... Hermite found similarly other expressions for F(2) +F(6) +... + F(4n +2), such as (1a) >( _1)t'0-1'+T(a-1) +2>(_] [4n+2-4a j -b where a and b range over all odd positive integers satisfying 4 + - - b2 0. 4n+2-a 2- > O. By means of two other formulas of Kronecker, Hermite evaluated similarly F(1) +F(5) +... +F(4n+l1), F(3) +F(11) +... +F(8n+3). He announced without proof that (2) F(3) +F(7) +... +F(4n +3) =2S [n c c' ] ' c>, c'>0 and satisfying (c+ 1) (2c+2c'+1) n +1, counting half of each term in which c'-O. T. J. Stieltjes165 stated that by the theory of elliptic functions he obtained the theorem: If d range over the odd divisors of n and ~ (n) =: ( _l) ) -l)+J(d2-L)= (-2), (0) - then, for n =2 (mod 4), in Eronecker's54 notation, P(n) =~~(n-2r2) =r (n -8r2), r=o, -+, +2,.... Thence he verified his16T earlier theorem (1) for the cases n=c2 and n=2k2 by the method used by Hurwitz in finding the number of decompositions of a square into the sum of five squares (see this History, Vol. II, 311). A. Berger,l66 to evaluate Dirichlet's14 series (2), namely, V= 1(k )o15 Comptes Rendus, Paris, 98, 1884, 663-664; Oeuvres, I, 1914, 360-1. 166 Nova Acta Regiae Soc. Sc. Upsaliensis, (3), 12, 1884-5, No. 7, 31 pp. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 137 A being a fundamental discriminant, started from Kronecker'sl71 identity (4a) in the form cA —=1 A\ - A\ ( -e)7i t - / ( VA)) ( ), where E is the sign of A, and c> 0. By separating the real from the imaginary and by a study of quadratic residues and non-residues, he obtained LX/2 /A. 2hkcr A (1) A <,: T^ sin - -_V -A., k>0. h=1 _- k Since (cf. Dirichlet14) (2) >-o sin nu r - 0 <2 n=1 n 2 <<U<27r, we get, by dividing (1) by kc and summing, Dirichlet's23 formula (6) for A<0. Similarly by the use of the identity -log (2 sin 2 = c nu n=-l Berger obtained Dirichlet's23 closed formula (8), for A>0. To obtain Dirichlet's23 second closed form, Berger took, for A<0 (cf. Dirichlet, Zahlentheorie, ~ 89, ed. 4, p. 224) r L \P/V i k \2/-l//-l' where17 r=1 ---(A/2), and p ranges over all odd positive primes. By means of (1), this becomes /A\1 2hF2 < -A/2 /A\Too -1 br (l)- -- h= /1 -) 2-' 1sin 2h(2k- 1) - r/A. But (2) implies that the final factor is 7r/4. Hence we get Dirichlet's23 classic formula (5). By parallel procedure, Berger obtained, for A>0, (3) V 2 <A/2 (A g cotih Cf. Dirichlet,23 (8). A. Hurwitz'67 gave without proof168 thirteen class-number relations of the 11th grade which he had deduced by the method which he had used to obtain relations of the 7th grade.169 For example, 6:HI(4n- K2)= (n) +, (n) +~ 2(n)-i(n), (i)=l where K ranges over all positive integers whose square is -n (mod 11); while 67 Berichte Sichs. Gesells., Math-Phys. Classe, 36, 1884, 193-197. 168 For proof, see F. Klein and R. Fricke; Vorlesungen iiber Elliptischen Functionen,217 Leipzig, II, 1892, 663-664. 169 Math. Annalen,184 25, 1885, 157-196. HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI +(n) =iax, where x ranges over those solutions of 4n=x2+ 11yl2 in which x and y are not 0, and (x/11) = -1; )2 () = [5Z(n)- 12(n)], () =I Z[3ZO (n)- Z(n)], in which Z (n) denotes the number of solutions of 4n= x2 + lly2 +z2 + lu2 for which x+y is even; Z,(n), the number for which one of x, z, x-z, x+z is divisible by 11. By eliminating ~2 and /3 from his set, Hurwitz obtained a new set which he showed to include J. Gierster's170 class-number relations of grade 11. L. Kronecker,171 unlike Gauss, studied quadratic forms ax2 + bxy+ cy2 in which b may be even or uneven. He defined primitive forms as those in which a, b, c have no common factor. He denoted by K(D) the number of primitive classes of discriminant D =b2 - 4ac. He put nn^- (D 1 D 2g D H (D)- = ~ - W ( h l h n l P if h=2h', h' uneven, in which the symbols of the last right member are the JacobiLegendre signs. Dirichlet's20 fundamental formula (2) is specialized as follows: (1), () ( ) F(h1) 2 ()F (amnl bmnn+cn2) *h, k\ a, b,c m,n v n where h, k range over all positive integers; m, n over all integers not both zero; a, b, c over the coefficients of a system of representative forms (a, b, c) of the primitive classes of the discriminant D=Do0Q2 (Do being fundamental); a>0 is relatively prime to Q; and b and c are divisible by all the prime divisors of Q; F(x) is any function for which the series in each member is convergent. By Dirichlet's methods (Zahlentheorie, Arts. 93-98) are obtained the following results: (2) rH(D) = K(D), D<O; H(D) = — D logT+ D>0. These are combined into one formula H(D)=K(D)J dz-, v~o)=g~u~j~) T2 -D where T, U denote that fundamental solution of T2-DU2=1 or 4 for which T/U is the greater. This is equivalent to (3) H(D)= (D) logE((D) E(D)= (T+UVD), r=l or 2. VD r But (cf. Dirichlet, Zahlentheorie, Art. 100), H(D)=H(Do)( l-(IDo )1), q q/ 170 Math. Annalen,148 22, 1883, 203-206. 171 Sitzungsber. Akad. Wiss. Berlin, 1885, II, 768-780. CHAP. VII BINARY QUADRATIC FORM CLASS NUMBER. 139 q ranging over the prime divisors of Q. Hence, (4) K(D) -1-f /A I log E o(D,) K(Do) { (q / q logE(D)' In the light of the identity (p. 780) (4a) (D) =_ (D)e2rcr I, c=1,3, 5,..., 2JDo,-1; r>0, \r I Do k\ (2) implies K (Do) 2Do k=l (, Do<0, (5) 1D ) k= lD \3 K(Do) log E (Do)=- ( — )log(1l-e2kri/Do), D>0O. H. Weber172 and J. de Seguierl73 have modified the above identity (4a) so as to be true also for Do=0 (mod 4), which is not the case in Kronecker's form of it. De Seguier has given the deduction in full of (5) and has shown that (52) holds also for Do<0. Dirichlett74 at this point needed to treat eight cases instead of Kronecker's two and de Seguier's one. Kronecker'75 had defined the function 0(g, o) by 0(g,.o) = -e~(V2w+4^-P)r, v=- l, -3, 5,..., and the function A by (', En, \ I2) (4A 2)e72(lT24+2)7ri. 0(q +T~1, )l) (a-r TW021 2) [0'(0, WI) o'(o, 2)] ' in which a, T are arbitrary complex numbers; o, o, are any complex numbers such that o1i and w2i have negative real parts. He176 found that if ow and - w2 are the roots of a+ bw+cw2 =, where b2- 4ac = -A is a negative discriminant, then /- A e2(2(ma+nr) 7ri (6) log (a, r,, 2) - lim o (am2+bmn+cn2)1+P 2 pr P=O m, n (a + bmn + and therefore A is a class invariant. Relation (6) was afterward developed by Kronecker177 into what J. de Seguier178 has called Kronecker's second fundamental formula. For DI, D' two arbitrary conjugate divisors of D=D'1 D2=-Do Q2 (1) is found to imply what J. de S6guier179 has called Kronecker's first fundamental formula, namely,180 (PIDQ2)(D2Q2)F(hk) h-1 kA —)( )f( ) h=l fc==l\ fl /F\ /T/ 2 a [( ) + (jA) mn(,)F(am2 +bmn+cn2), a, b, c V m, n M 172 Gotting. Nachr., 1893, 51-52. 73 Formes quadratiques et multiplication complexe,226 Berlin, 1894, 32. 174 Zahlentheorie, Art. 105, ed. 4, 1894, 274-5. 175 Sitzungsber. Akad. Wiss. Berlin, 1883, I, 497-498. 176 Ibid., 528. 17 Sitzungsber. Akad. Wiss. Berlin, 1889, I, 134, formula (16); 205, formula (18) 213 78 Formes quadratiques et multiplication complexe, 1894, 218, formula (3).226 17 Ibid., 133, formula (6).226 180 L. Kronecker, Sitzungsber. Akad. Wiss. Berlin, 1885, II, 779. 10 140 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI with ranges of summation as in (1), while 2am/n +b U U/T and n>O, if D is >0; A is an arbitrary number relatively prime to 2D and representable by (a, b, c). An elegant demonstration has been given by H. Weber.l81 Take Q = 1, D1<O, D> 0, F(x) =x-l-P. When (6) is applied to the right member, the result, when p=0, is (7) H H(D)H ((D2)= () log c [' (0, o)) O(0, o,)]-, -D. 'V a, b, c\ / This formula refers the problem of the class-number of a positive discriminant to that of a negative discriminant. For the purposes of calculation, this formula has been improved by J. de Seguier.l82 L. Kronecker183 considered solutions (U, V) of U2+DV2-=4p, where p_1 (mod D), D a prime=4n+3>0. If xP=1, aD=1, X / 1, a=# 1, and g is a primitive root of p, then (x a+ + axg2aX+... +a (P-2)axgP-2) = u +U - V -~D, a where a ranges over the incongruent quadratic residues of D, and u and v are integers. Whence finally he stated that U and V are determined from u+vV-D _ /U~ IVL-D Db-a u-vV - D \- U-VV-D Cf. Dirichlet's23 formula (6). A. Hurwitz184 stated that his185 modular equations of the 8th grade134 yield those class-number relations which L. Kroneckerl24 had given in Monatsber., Berlin, 1875, 230-233. He modified Gierster'sl45 deduction of the class-number relation of the first grade by showing that a modular function f(J, J') has as many poles as zeros in the fundamental polygon. For genus138 p>0, Hurwitz employed a system of normalized integrals ji(o), j2((),..., jp(W) of the first kind on the Riemann surface formed from the fundamental polygon for the largest invariant sub-group of grade q. For arbitrary constants e, the 0 functions186 of jr have the property 09[jr (T(w) ) -er] =O[jr(o) — er]e, k 2tr(jr()) -er) - Ct, r=l where T is an arbitrary unit substitution - (?) (mod q); while t1, t2, t3,..., t, Ct depend only on T. Constants Cr are so chosen that o [j (,) -jr(Q) -Cr] =0[jr() - j(o) + Cr], and 0=0 when and only when the zero regarded as a value of o(and 0) is relatively 181 Reproduced by de Seguier, Formes quadratiques, 332-334. 182 Formes quadratiques et multiplication complexe, Berlin, 1894, 314, (25). 183 Gottingen Gelehrte Anzeigen; Nachrichten Konigl. Gesells. Wiss., 1885, 368-370, letter to Dirichlet. 184 Math. Annalen, 25, 1885, 157-196. 185 Gottingen Nachr., 1883, 350. 186 Cf. B. Riemann: Jour. fiir Math., 65, 1866, 120; Werke, 1892, 105; Oeuvres, 1898, Mem. XI. 207; C. Neumann, Theorie der Abel'schen Integrale, Leipzig, 1884, Chaps. XII, XIII. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 141 equivalent to 2Q,, 0, *.., (op- (and ), op, (op,.p+., * 2P-2), where o, o)2,..., to2P-2 are constants chosen almost'87 arbitrarily; moreover, that zero is of the first order. The transformations R, (o), R2(o),... are a system of representative substitutions188 of order n and are a+b (mnod q), where a, b, c, d are fixed for all R's. Consider the function I (o) = L [jr () -jr (Ri (o) )-c,], i where if n is a square, we omit the representative n(mod q), Vn which is relativelyT34 equivalent to o. Aside from the zero values which are due to the choice of oi, 0w,., 2-2, and aside from the rational points o, the theory of the zeros189 of a 0-function shows that, since ~((o) is reproduced except for a finite exponential factor under the substitution T(cw), S(o) vanishes in the fundamental polygon as many times as there are identities ~ /ab d\ - Ud' c -b'c'-b =n, (-.d- cd) (mod q)* From this point Hurwitz treats the 0-functions as Giersterl45 had treated the factors q(o) )-v(/) of the modular equation and his determination of Gierster's a differs only in details from Gierster's determination. To complete Gierster's nine class-number relations'90 of the 7th grade for n E 0 (mod 7) and without recourse to non-invariant subgroups, Hurwitz, after F. Klein,l91 put Z ((o) =- ( —1) "Vq[7(2v+1)-+8]2, ((o) = (-1) V"q'[7(2v+l)+1]2 Z4 () = (-1 ) q [7(2+l) +2]2. Three normalized integrals of the first kind and of grade 7 are I(o) = - Ix, z.6 (Olq) dq ',. (m) q2 r= /, 1 4; summed for values of m r (mod 7), where necessarily,r(m) =-1a, the summation extending over all positive and negative integer solutions a, /f of 4m = a2 -79", m-r (mod 7), (a/7) =1. Now Ir(o) has the property I(n) I,. (R (o)) =const., or / (n) r,(S ()) + const., i=1 according as (n/7) = -1 or + 1, while S()a + b (mod 7), cow +d 187 Cf. H. Poincare and E. Picard, Comptes Rendus, Paris, 97, 1883, 1284. 18s F. Klein, Math. Annalen, 14, 1879, 161. 189 B. Riemann, Jour. fir Math., 65, 1866, 161-172; Werke, 1892, 212-224. o90 Math. Annalen, 17, 1880, 82; 22, 1883, 201-202. 191 Ibid., 17, 1880, 569. 142 HISTORY OF THE THEORY OF NUMBERS. [CHCAP. VI and (n)=-=1a, the summation extending over all positive and negative integer solutions a, f/ of 4n=a2 +7 /2, (a/7)=1. Let this property of the integrals Ir be possessed by the integrals j, j2, j,. Hurwitz put 4(n) ((/,O)) <o= n [jr (o/) - jr (Ri (o ) )-Cr], i=1 (, f0[jr(o') -jr(S(o)) -Cr]-'P(n), if (n/7) =1, (vl/( t^)=-A 1 ~, if (n/7)=-1, ~()' -(, t>) -(J, O ). (o', )), F(o/, () (= (o";) where too, wo are arbitrary fixed values of o with positive imaginary parts. Then F( w, ) is invariant under T(o) and hence as a function of o and of w' is an algebraic function belongs to the Riemann surface of the 7th grade. F (w, o) =0 expresses algebraically the modular correspondence'92 of grade q and order n. F((', () is an algebraic function which belongs to the surface and has as many zeros as poles in the fundamental polygon. Hence (1) a-k.-(n) -=2(n)-2i(n) if (a ) - (v P), where k is the number of zeros of 0[jjr(o)-jr(S(w))-cr] in the fundamental polygon, and o has the value given by Gierster.l45 Similarly (2) a=2 )(n) -6((n) +r, if ( ) - (v 0), where - =4 or 0 according as n is or is not a square. From (1) and (2) and the relationl45 2(Up+U2+ U4p) = (n) —(n), p n.( )n Gierster'sl93 class-number relations of grade 7 follow at once; for, Gierster's'39 (n) is Hurwitz's - 2 (n). A. Hurwitz'94 generalized completely his184 deduction of the class-number relations of grade 7 to grade q, where q is a prime > 5; and showed that the right member of these relations is 24 (n) plus a simple linear combination of coefficients ~ (n) which occur in an expansion of Abelian integrals of the first kind and of grade q. That is, if a(n) be determined in terms of class-number as by Giersterl45 and Hurwitz,'84 o(n) -2< (n) - rh= ( ^)+ h0(n) +... + h (n), where,=2(p-1) or 0 according as n is or is not a square; and h,, h2,..., hg are independent of n. Klein and Fricke217 have since shown for q= 7, 11, how the h's may be simply evaluated when the O's are known. 192 Cf. A. Hurwitz, Gottingen Nachr., 1883, 359. 193 Math. Annalen, 22, 1883, 199-203 (Gierster 48). 194 Berichte K6nigl. Sachs. Gesells., Leipzig, 37, 1885, 222-240. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 143 E. Pfeiffer195 wrote H(n) for the number of classes of forms of negative determinant -n, and sharpened Merten's17 asymptotic expression for the sum:H(n) to the equivalent of (n) = 9- -+ (X+e), n=1 9 ( where the order"7 only of the last term is indicated and e is a small positive quantity. Pfeiffer, in a discussion which lacks rigor, indicated a method of proof (see Landau330 and Hermite204). L. Gegenbauer198 denoted by f(n) the number of representations of n as the sum of two squares, and deduced from four of Kronecker's formulas like124 (1) four formulas similar to and including the following: 12 S E(x)=f2(n))+2 f(n-2), x=1 X=1 where54 [Ir] E(n)= 2F(n)-G(n), f,(r) = f(x) = [V r-x 21. x1 X-=0 His earlier result'97 [Vm/a] ____ S [Vm-ax21 =- +O ( m) 1=0 4V\/ transforms this into n S E(x) -1T3/2+ 0 (n). x=1 (For the notation 0, see F. Mertens.l7) The other analogous results are lim F(4x+a)/n3/2n =, lim E F(8x+3)/n7 3/2=irV20 n=cao $=0?i==c x=O where a=l or 2. Hence the asymptotic median number in the three cases is 7rr/n, ~7rVin, 7rVn/2. These four results combined with those of Gaussl98 and Mertensll7 give the asymptotic median number of odd classes as 7rVn 1 + 1 1)} (3)-+ 3+ 33+1 1+ T2 7(i(3,> ~ + T,) 43 Gegenbauer99 derived from four of Kronecker's200 and four of Hurwitz's202 formulas, twelve class-number relations with more elegance than he'96 or Hermitel64 had derived three of the same formulas. For example, from the following formula of Hurwitz,202 4 F(8n+l)q "'2n +-0(q)2(), n-== 195 Jahresbericht der Pfeiffer'schen Lehr-und Erziehungs-Anstalt, Jena, 1885-1886, 1-21. 196 Sitzungsber. Akad. Wiss. Wien, Math-Natur., 92, II, 1885, 1307-1316. 197 Ibid., 384. 198 Disq. Arith.,4 Art. 302; Werke, II, 1876, 284. 199 Sitzungsber. Akad. Wiss. Wien., Math.-Natur., 93, II, 1886, 54-61. 200 Monatsber. Akad. Wiss. Berlin,124 1875, 229. 144 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI it follows that 4F(8n+1) is the number of integral (positive, negative or zero) solutions of 8n 1 = 82 +8y2 + (2z 1) 2 Put X2 + y2 = and solve for z. For a fixed 1c, the number of integer values of z as n ranges from 1 to N is therefore [-1- V8 N+ -8i c -I j]. Hence An n 2 S F(8x+ 1)= E f(x) [V8n + -8 +~l], =o0 '=o where f(x) denotes the number of representations of x as the sum of two squares. The symbol f(x) is decomposed so that the last formula becomes n S PF(Sx+l)=1-7rV/2n3/2 + o (n), X=O with 0 as in Mertens.117 As in the previous case,196 Gegenbauer now finds that the asymptotic median number of odd classes of the determinant - (8n+ i), i=1, 2, 3, 5, or 6 is 7rV/n/2. Gegenbauer201 without giving proofs supplemented his earlier listl99 of 12 classnumber relations with 20 others which are easily deduced by processes analogous to those used before'99 and which include the following three types: 5 F(8x+3) [V8/8n+1-8x+~]=- S,(2x+l), aX=O x=0 in which [presumably] 1 (n) denotes the number of representations of 4n as the sum of four uneven squares, where the order of terms is regarded, but (-a)2 is regarded as the same as (+a)2. n n S F(16ix+14) 2 E p(8sx+))[-V4n +1-4x —1], x —0.-=0 p(m) = ( - 2/d,), d, ranging over the odd divisors of n. F (8x+6) =21(-1)y-1+4 (-1).- 2(n- 2y ) - z + 1=0 y y, L 8y-4 y > 1; z > 1; 2n-4y2+44y-zx2+Z 0. A. Hurwitz202 employed four formulas of Kronecker203 all of the same type and including (1) 4SF(4 + 2) q=q-:02 (q)03(q), (2) 4: (4n +l ) qn=-Iq4-2 (q) *3 (q). He enlarged the list to 12 such formulas by simple methods, for example by replacing q by -q in (1), adding the result to (1), and then using the relation o2(q) =202(q2)03(q2). 201 Sitzungsber. Akad. Wiss. Wien, Math-Natur., 93, II, 1886, 288-290. 202 Jour. fir. Math., 99, 1886, 165-168; letter to Kronecker, 1885. 203 Monatsber. Akad. Wiss. Berlin,124 1875, 229-230. CHAP. VI] BINARY QUADRATIC FoORM CLASS NUMBER. The result in this case is (5) 2~F(8n+2) q= q9O2(q)03(q)03(q2). Seven class-number relations are obtained similarly to the following. We multiply (2) by 2 (iql). The relation 0(iq)6 (q) =0-{ (iq) now gives 402 (iq') F(4n+ 1) q+C= O2(q)O' (iq4); and the equating of coefficients here gives,h(-) ('-'/P s(m -h2)=-jQ(n), m =3 (mod 8) in which h is uneven and positive, QO(m) = (-2/v)v, where v ranges over all positive uneven numbers satisfying m =v + 2n2. C. Hermite204 represented the totality of reduced unambiguous quadratic forms of negative determinant and positive middle coefficient by (2s-+r, s, 2s+r+t), r, s, t=1, 2, 3,.... Hence in S = 2:q(28r) (2s+r+t) -s2 the coefficient of qN is the number of unambiguous classes of determinant -N. And if we put n=2s + r, we get 2 -!-"-2 n=3, 4, 5,...; s-=,3,..., [ The number of ambiguous forms (A, O, C), A - C, of determinant -N is the number of factorizations N= n(n+ i), where n is a positive integer and i > 0. This implies that the number of ambiguous forms of this type is the coefficient of qN in the doubly infinite sum S.-:= q"(t ' w -t q n,i 1q -- q' Similarly the number of ambiguous reduced forms of the type (2B, B, C) and (A, B, A) of determinant -N is the coefficient of qN in the expansion of,t2~2n S=!- 1 t- q' n=1, 2, 3,.... 1- q2 -This gives205 qfl q, 2n n= 1, 23,...; 1 + 2- 2- q12 _1 _ ql2n m=1,3, 5, 7.... Hence, if H(n) denotes the number of classes of determinant -n, 4n2 n22-I21 2qn2an-s2 ff(~ )q=2 — q +S qn+2 -_qn. i —qn 1 - q n 1-qq 204 Bull. des sc. math., 10, I, 1886, 23-30; Oeuvres, IV, 1917, 215-222. 205 Cf. C. G. J. Jacobi, Fundamenta Nova, 1829, Art. 65, p. 187; Werke, I, 1881, 239 (transformation of C. Clausen). 146 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI We divide each member by 1-q and expand according to increasing powers of q. Then the coefficient of qN in the left member is U H(1) + H (2) +... +H(N). By the use of the identity206 qb (N+a-b ) (1) ( 1-q)(i q) a the coefficient of qN in the second member becomes U= E (N+ n-n) 2+ E(N-n2 +Wp:) + V (N -2 ) n / 2n n Neglecting quantities of the order of E (VN) =v, we get.E(N fn-n2 )= N n-n 2 =N (li+ +... )_2 v -E(:n) N N J(i+~~+... + j -. - rZ, v, 2 = (N(- log N + ) -N; E (-n ) N-n 2 N (1 V+ + + + _. — v 2n- 2n 2 2 =IN(~ log N+C7) —iN, where C is the Euler constant.830 In short, U=3NlogN+ 2 N +s2 n2 Geometric207 considerations give the approximate value of the last term as 2 IJN+X - Ydxdy, x,y>O, where the limits of integration are given by the relations y>2x, N+x2-y2>0. Hence for N very great, U=2 rNg. Cf. Pfeiffer,195 Landau.330 L. Gegenbauer,208 employing the same notation as had G. L. Dirichlet209 and the same restrictions, obtained by new methods the results of Dirichlet, that the mean number of representations of a single positive integer by a system of representative forms of fundamental discriminant A is T (-, - if A>0; 27rK(A)/V-A, if A<0, 1== \ x I < where K (A) is the number of classes of negative discriminant A. For example, in the first case, the identity -~ - -~ )= ( -- nS -L /==. (14~\?I n x=l x, y= S y r= r d d in which, presumably, e(x) =0 or 1 according as x <1 or >1; and the last summation extends over divisors of r, implies that -l a(\d n A-1/A\ X=l 12 / X= l \ \ L 206 C. Hermite, Acta Math., 5, 1884-5, 311; Oeuvres, IV, 1917, 152. 207 Cf. R. Lipschitz,102 Sitzungsber. Akad. Wiss. Berlin, 1865, 174-175. 208 Sitzungsber. Akad. Wiss. Wien, 96, II, 1887, 476-488. 209 Zahlentheorie, Braunchweig, 1894, 229; Dirichlet.19 CHAP. VII BINARY QUADRATIC FORM CLASS NUMBER. 147 where d ranges over the divisors of x and T (A/d) is Dirichlet's93 expression (2) for the number of representations of x by a system of representative forms of determinant A. Hence - I nx - n W ^UT T n -TnS T ^U- - 7; +T-: + ~:( 1 d (d ) x=1 ) XX n X x =l \x X=[Vn]+l X where 0 < Ec<1, and each of the last three terms remains finite when n becomes infinite. Gegenbauer210 defined a certain function by d dk \ dJ) in which (A/d) is the Jacobi-Legendre symbol, d ranges over the divisors of n, and z (x) is the Moebius function (this History, Vol. I, Ch. XIX). Then F [] ( Xk )X (X)= ()z z=,L l X x=l X if A is prime to 1 2, 3, 3..., n. This relation combined with Kronecker's71 formulas (2) and (5) gives the number of classes of a prime discriminant A. That is, g Ia)= r AInr —1 A1-1z~Aj K ^(A) =, [ x I ( Xl(X), K(A) - (2 - ( For example, if A= —, Xo(1) =, Xo(2) =0, Xo(3) =2, XI(1)=1, XI(2)=1, Xi(3) =4, xi(4) =2, Xi(5) =6, Xi(6) =4. Therefore K( — ) =1. C. Hermite211 employed an earlier result69 l_ q = F (n) q- n, n=4m —1,, a-. ql(af2~) -c~2 r= -I a-1,3,5,... ( for a= 2c' + 1, divided by q1, then applied his204 identity (1), and equated coefficients of qm-1 and obtained F (4r- 1)-d + d' ) where d, d' are of the same parity; d' ( d; m (d + 1) (d'+ 1); and the coefficient 2 is to be replaced by 1 if d=d'. But when in mathematical induction m-1 is replaced by m, the right member of the last equation is increased by double the number of solutions of - dd'1 =c, or 4m-1=4(c+d) (c+d') -(2c-1)2, in which c=1, 2,..., m; d d' (mod 2), d'>d; while, if d'=d, each solution is counted ~. This gives the value of F(4m -1). 210 Sitzungsber. Akad. Wiss. Wien (Math.), 96, II, 1887, 607-613. 21 Jour. fur Math., 100, 1887, 51-65; Oeuvres, IV, 1917, 223-239. 148 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Hermite equated the coefficients of certain powers of q in two expansions of Hi (0) and found that, for m-3 (mod 8), the number of odd classes of the negative determinant -m is S4((m-b2), in which b=O, ~2, +4,...; b2<m; and >(m)= (_-1)(d'+l), d' ranging over the divisors of m which are > Vm and =3 (mod 4). P. Nazimow212 gave an account of the use54, 145 of modular equations, and of Hermite's69 method of equating coefficients in the theta-function expansions, to obtain class-number relations. X. Stouffl28 of Ch. I extended Dirichlet'sl9 determination of the class-number when the quadratic forms and the definition of equivalence both relate to a fixed set of integers called modules. L. Kronecker213 let ax2+ bxy+ cy2 be a representative form of negative discriminant D-= -A=b2-4ac; put a=a0o \/ and (Cf. Kronecker'75) A (0, 0, (01) 0()2) — 0'(0, 21 ) ((~<>- -Co - )2 2 _ -4-V 0 c1 02 2C He obtained the fundamental formula pl= -[ + 2r (am2+bmncn2 ) =log 47r2+2C-log A' (0, 0,,,1 ), p=O P 7rn. (a rr^- bmn -cn2)1\ where C is a constant independent of D, a, b, c. When each member of this identity is summed for the K(Do) representative forms of fundamental discriminant D,, the result enables Kronecker171 to evaluate the ratio H'( —Ao)/(-Ao) in terms of K(Do), where k=(- 1) k k=1 k k This is called Kronecker's limit ratio. H. Weber214 denoted by o the principal root of a reduced quadratic form of determinant -m, and denoted by j(w) the product of F. Klein's134 class-invariant J by 1728. The class equation (1) II[u-j(W)] =o, in which w ranges over the principal roots of a representative system of primitive quadratic forms of determinant -m, he expressed by (2) H,,(u)=0, or (3) ' (u) =0, according as the forms are of proper or improper order. By applying transformations of the second order to a, he set up a correspondence between the roots of (2) and (3). This correspondence is 1 to 1, if m = -1 (mod 8); 3 to 1, if mn 3 (mod 8), except when m= 3. Whence he obtained Dirichlet's20 ratio between h (D) and h'(D), D<0. 212 On the applications of the theory of elliptic functions to the theory of numbers, 1885, (Russian). Summary in Annales Sc. de l'Ecole Norm. Sup., (3), 5, 1888, 23-48, 147 -176 (French). 213 Sitzungsber. Akad. Berlin, 1889, I, 199-220. 214 Elliptische Functionen und Algebraische Zahlen, 1891, 338-344. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 149 Weber215 gave the name (cf. Dedekind's128 valence equation (1)) invariant equation to (4) l[it) ) a+bO )] of order ad-bc=n, in which the g.c.d. of a, b, c, d is 1, and,_ c + dw a+bw is a complete set of non-equivalent representatives. He observed that, if W furnishes a root j((o) of (4), then o must be the principal root of a quadratic form (5) A(2 +B +C, B - 4A C-D, where, for a positive integer x, b =Ax, c= -Cx, a- d =Bx; and if we set a+ d= y, we must have (6) 4n=y2 - Dx2. Conversely, for each of the k representations of -D in the form -D= 4n-y2 x2, there are CI(D) = h'(D) forms (5) each of whose principal roots furnishes one root of (4). Hence (4) can be written (cf. Weber's Algebra, III, 1908, 421) (7) CHI (u)H2(U)..=0 u=j(&). If j(o) is a root of (4), expansion of the left member in powers of q=e7iw shows that the degree of (4) in j(w) is 2-e +(e) ++V(^) or 2 8 +(e), e e according as n is or is not a square (cf. Dedekind,128 (2)) where a> Vn is a divisor of n. The degree of (7) in j(w) is:h'(Di)ki, summed for i=l, 2, 3,.... For brevity, (4) is written Fn (u, u) =0. The simplest case of deducing a classnumber relation of L. Kronecker's type48 is presented by equating two valuations of the highest degree of u=j(o) in the reducible invariant equation F,,n (u, U) * F7w, (u, u).Fn'2 (u, U)...0, where ns, n', nh,... are derived from n in every possible way by removing square divisors including 1, but excluding n when n is square. The relation is K(n) +2K(n-1) +2K(n-4) +.. +2K'(4n-1) +2K'(4n-9) +... = 2a or 2:+a+Vn+, according as n is not or is a square. Here K(m) denotes the number of classes of 215 Elliptische Functionen und Algebraische Zahlen, 1891, 393-401. 150 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI determinant - m, and K' (m) denotes the number of classes of determinant -m derived from improperly primitive classes. Finally, S3 is the sum of the divisors of n which are > Vn. J. Hacks216 considered the negative prime determinant -q, where q=4n+3; he put ~(q-i1) acs-1- s2 S= ~ [-, S- I --, si i s=1 L g J and found that the number of properly primitive classes of determinant -q is h=j-(q-1)-2S'+4S. This is given the two following modified forms a - -(q —3) h=- - 2 (-1)[V2q.s], 2 1 -1 (q —3) [2s2/q] h 2= +2 a (-+); 2 ss=l 1 and finally is reduced to Dirichlet's23 formula (6). F. Klein and R. Fricke217 reproduced the theory of modular functions of Dedekind128 and Klein/,34, 138 also (Vol. II, pp. 160-235, 519-666) the application by Gierster,135, 139, 145, 147 Hurwitz,167, 184, 194 and Weber214 of that theory to the deduction of class-number relations of negative determinants. They gave (Vol. II, p. 234) the relations of grade 3 which come from the tetrahedron equation and (Vol. II, pp. 231-233) the relations of grade 5 that come from the icosahedron equation. Their formulas (1,) p. 231, and (7), p. 233, should all have their right members divided by 2. They reproduced (Vol. II, pp. 165-73, 204-7) the theory of the relation between modular equations and Smith's132 reduced forms of positive determinant. In connection with Hurwitz's194 general class-number relation of prime grade q> 5 and relatively prime to n, Klein and Fricke constructed a table of values of Ai! and Xi for n < 43. A sample of the table follows (p. 616): n I 1 1 2 1 3 I2 I | XI X2 1 -1 1 0 2 0-1 3 1 1 1 6- 1 0 4 2 0 1 7 1- 0 For q= 11 and (n/q) — 1, Hurwitz's first general formula becomes 6 E H(4n- K2)=2 (n) + t1Xl + t2X2, 3V —n where K is positive or negative and K2 (3V-n)2 (mod q). Hence, by the table for n=2, -=6, 12H(4) =6-t2, 1211(23)= 24+t,. But it is known that H (4) =-I; H(23) =3. Therefore t = 12; t2=0. G. B. Mathews218 reproduced in outline the researches of G. L. Dirichlet93 on the number of properly primitive classes of a given determinant; and those of Lipschitz219 on the ratio of the numbers of classes of different orders of the same determinant. 216 Acta Math.. 14, 1890-1, 321-328. 217 Elliptische Modulfunctionen, I, 1890, 163-416; II, 1892, 37-159. 218 Theory of Numbers, Cambridge, 1892, 230-256. 219 Ibid, 159-170. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 151 H. Weber,220 by arithmetical processes, obtained L. Kronecker's221 expression for the number h of primitive classes of forms ax2+bxy + cy2 of discriminant D. For D-=Q2.A, A a fundamental discriminant, he obtained by Dirichlet's20 methods, Kronecker'sl71 ratio (4) of the class-number of D and of A. By the use of Gauss sums, he transformed the former result for Q-1 into (1) 17i- 2' (A,s)s, A<0, (2) h log i(T+ UVA) = -S(A, s)log sin S7r/A, A>0, S in which222 (A, s) is the generalized symbol (A/s) of Kronecker171; and 0<s< ~A. By Dirichlet's methods, he obtained the analogue of Dirichlet's23 formulas (5). See Lerch,240 (4). By use of the Gauss function () = lim (log m- --- - m=oo u+l u+ 2 ' ''u+m ' the formulas written above become (3) h - - ( (,v) cot s,,<0; (4) hlog (T+ UVA/)= - (Av) [( ( -+ -)] A >0 O<v< ~A/2 (cf. Lebesgue,36 (1)). For A= -mn <0 and uneven, (3) is equivalent (cf. M. Lerch,238 (1)) to (5) -- cot: ~ Weber transformed (p. 264) his formula (2) above by cyclotomic considerations223 and observed that h (A) is odd if A is an odd prime or 8, and even in all other cases. (Cf. Dirichlet, Zahlentheorie, 1894, ~~ 107-109.) P. Bachmann224 reproduced (pp. 89-145, 188-227) a great part of the classnumber theory of Gauss4' 9 Dirichlet,93 and (pp. 228-231) Schemmel95; and also (pp. 437-65) the researches of Lipschitzl02 and Mertens'17 on the asymptotic value of h (D). J. de Seguier225 showed that Kronecker's171 formula (52) is valid for Do<0, if in the right member, Do be replaced by IDo. This proof is reproduced in his226 treatise. J. de Seguier226 wrote a treatise on binary quadratic forms from Kronecker'sl71 later point of view making special reference227 to two fundamental formulas of 220 G6ttingen Nachr., 1893, 138-147, 263-4. 221 Sitzungsber.171 Akad. Wiss. Berlin, 1885, II, 771. 222 Cf. H. Weber, Algebra, III, 1908, ~ 85, pp. 322-328. 223 Cf. Dirichlet,93 (1); Arndt.53 224 Zahlentheorie, II, Die Analytische Zahlentheorie, Leipzig, 1894. 225 Comptes Rendus, Paris, 118, 1894, 1407-9. 226Formes quadratiques et multiplication complexe; deux formules fondamentales d'apres Kronecker, Berlin, 1894. 227 Ibid., 133, formula (6); p. 218, formula (3). 152 HISTORY OP THE THEORY OF NUMBERS. [CHAP. VI IKronecker.228 He extended (p. 32) Kronecker's171 identity (4a) in Gauss sums (cf. H. Weber, Gott. Nachr., 1893, 51) to the form D0-1 ) e2s^o =-hi 0 ( o_ 1)-(sgn h-1) (gn DO-1) hO, where sgn x= +1 or -1, according as x is > or x<0, while Do is a fundamental discriminant. Then, whether D is positive or negative, it follows at once in Kronecker'sl71 notation that the number of primitive classes is given by (1) K(D) log E(Do) = VDoH(Do) = Do:( n - 1n=l1-1 ' IDol-1/D oo e2nkri/lDolj D,1o-1 i== c S =3. (^ ^e'^^^log (1_ e2k(li/D)lo), k=l n=l k=1 in which E(Do) is a fundamental unit; and, if z=rei3, then logz=logr+i0, -7r<0<7r (pp. 118-126). For Do>O, this formula is Kronecker's171 (52). Elsewhere de Seguier225 repeated briefly his own deduction of (1). By noting that log (1-e /Drol) =log 2 sin W/|DO +~(i-7r-c7r/|Iol), he obtained from (1) two distinct formulas; one being Kronecker's171 (5a) and the other (p. 127) being Weber's220 (2), 1 Do D)10 111 7 Dr K(1D) l - ) (Do)log in D > 0. By a study of groups of classes in respect to composition of classes, de Seguier (pp. 77-96) obtained the ratio of Cl(D.S2) to CI(D). Cf. Gauss,4 Arts. 254-256. Denoting the Moebius function (see this History, Vol I, Ch. XIX) by e,,, de S6guier found (p. 116) that for any function F which insures convergence in each member of the following formula, we have Q ( F(1)=F Ez ~, F(nd). m=l1 m dIQ n=1 If a, b, c are arbitrary constants (eventually integers) and F is taken such that F(xy) =F(x).F(y), we have F 2 )(am2+m - + cn2) = ea F(d)F(adm2 +bmn- + - n2), m,n M dlQ m, n d m, n=, ~1, 2,..., ~oo, except m=n=O. Let F(u) be p/u1+P. Since, for such a function, lim S F(am2 +bmn+cn'2) p —O m, n 228 L. Kronecker,171, 213 Sitzungsber. Akad. Wiss. Berlin, 1885, II, 779; 1889, I, 205. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 153 depends only on b2 - 4ac, we have lim p S (aom+bmn+ cn2)-l-p =(Q) lim p(adm2+bmn+ n2-)-1-P p=O m,nn Q p=O d But'71 TrH(D) =K(D)lim pS (am2+bmn+cn2)-1-p p=O m, n Hence we have, for D<0, H (DoQ2) p(Q) es H(Dod'2) d 4 TDOQ2 TDOW2 (dd4). (4) D (Do Q2) Q - d2K(Dd2)Dd (DodQ) To this formula is applied the following lemma due to Kronecker229: Let f(n), g(n) be two arbitrary functions of n and let h (n) = f(d)g(d') (dd'=n), and let g have the property g(mn) =g(m)g(n), g(1) =1; then f(n)=-Edg(d) 'h(d') (dd'=n). Hence we deduce from (4) the new relation (p. 128) H(DOQ2) Q- K p(d) H(Dod2) D<O. TDQ (DQ)2) dd=QDod2 d' K (Dod2) ' For discriminants D,<0, D2>0, de Seguier gave the following approximation formula (p. 314): K(D) log E (D,)= 2K(D2) log a+), 2K(DO) A 6 the summation extending over a system of primitive forms (a, b, c) of discriminant D=D,.D2; while A is an arbitrary number representable by (a, b, c) and relatively prime to 2D. M. Lerch,230 in the case of Kronecker's forms of negative fundamental discriminant -A 5 (mod 8), gave to Dirichlet's20 equation (2) the form 2 S' F(amn2+bmn+cn2)= S (-A/h)F(hkc), a, b, c m, n h, k m, n=O, ~1, ~2,..., except n=n= 0; h, k=1, 2, 3,.... He took F(x) =(-1)j e-X7*'l\A and obtained (1) / W (-1)S/ _e 1m,+m-n Q- (ram'+bn+cn2)/V-: (V -( 1)_t1c-"ker/1. a, b, c mn h, k h But by taking a(=r=0 in Kronecker's23T fundamental formula, it is seen that the left member of (1) would vanish if it contained the terms with m=n=0. Hence the left member of (1) is - Cl( -A), and (1) can be written h ^ ) (-1) h" i = H - ( —1) lb-e-h~/va. 229 De Seguier's Formes quadratiques, 114; L. Kronecker, Sitzungsber. Akad. Wiss. Berlin, 1886, II, 708. 230 Comptes Rendus, Paris, 121, 1895, 879. 231 Sitzungsber. Akad. Wiss. Berlin, 1883, I, 505.175 154 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI By expressing the right member in terms of a 0-function,175 we obtain o p -2 ) p )i YC01(-A) (A +iVA) ri, =A A+tA-2 y=- if A=3; yl if A>3. G. Osborn,232 from Dirichlet's23 formulas (6) and his own elementary theorems233 on the distribution of quadratic residues, drew the immediate conclusion that the number of properly primitive classes of determinant -N, N a prime, is -1 (N-)- (R), 1~ 8m '- >0, but is 3 times that number if N= 8m + 3>0, where S (R) is the sum of the quadratic residues of N between 0 and N. *R. Gotting234 found transformations of the more complicated of Dirichlet's23 closed expressions for class-numbers of negative determinants. A. Hurwitz235 denoted by h(D) the number of classes of properly primitive positive forms of negative determinant -D. Let p be a prime =3 (mod 4) and write p'= (p -1). Since (s/p) s' (mod p), Dirichlet's25 result (5) implies h (p) p'+P'+ 2.. + p+p' (mod p). The right member is the coefficient of (1) ( —l)IW-1)xP'/p'! in the expansion of cos Ix - Cosjpx (x) =sin x+sin 2x+... +sin p'x= cos cos p g sin ~x This numerator is congruent to cos 2x-1 modulo p, and by applying a theorem on the congruence of infinite series, we get (x cos -1 -2 sin2 - tan (mod p) 2 sin x - 4 sin ~x cos x - x (m But when x is replaced by 4x, (1) is multiplied by 41' or 2P-1 =1 (mod p). Hence h(p) is congruent modulo p to the coefficient of (1) in the expansion of -i tan x. When p l (mod 4), we employ the expansion of - sec x. Other such theorems give h(2p). The same result of Dirichlet is used to prove that if q 1 (mod 4) and q has no square factor >1, and if q-I 1{( —) sin x - () sin 3x + ()sin 5x-.. sin(q-2)x} cos q q q q = cRx + C2X3/3! + 3X5/5! +... 232 Messenger Math., 25, 1895, 157. 233 Ibid., 45. 234 Program No. 257 of the Gymnasium of Turgau, 1895. 235 Acta Math., 19, 1895, 351-384. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 155 and if p 3 (mod 4) is a prime not dividing q, then h(pq) -(-1l)(P+1)C_(p-+_) (mod p). There are analogous theorems for h (pq) and h (2pq) for all combinations of residues 1 and 3 (mod 4) of p and q. To obtain a lower bound for the number of times that 2 may occur as a divisor of h, the number of genera of the properly primitive order is calculated.236 If hg(D) denote the number of classes in a properly primitive genus of determinant -D, the parities of hg(pq) and hg(2pq) depend only on the values of (p/q) and p (mod 8) and q (mod 8), and are shown in tables. By combining the two theories of this memoir one obtains, for special q, results such as the following: If p- 3 (mod 4), h(5p) is the least positive residue modulo 2p of (-1)'(P+1)'c(p+), where c1, c2,... are the coefficients in the expansion sin x +sin 3x =3 x2n1 -cos 5x 3! (2n-1)! F. Mertens237 completed the solution of Gauss' problem (Disq. Arith.4, Art. 256) to find by the composition of forms the ratio of the number of the properly primitive classes of the determinant S2.D to that of D. He modified Gauss' procedure by taking schlicht forms (Mertens37 of Ch. III) as the representatives of classes and by means of them found for any determinant the number of primitive classes which when compounded with an arbitrary class of order S would produce an arbitrary class of order S (Mertens37 of Ch. III). M. Lerch238 rediscovered Lebesgue's36 class-number formula (1) above, and wrote it for the case A= = 4m + 3, a prime: 2S' cot 7r = 4i\/P Cl(-p), =l, 2,..., p-l, (-)=1. By replacing k by a2 -p[a2/p], he obtained Weber's formula220 (5): -- ~ (p' I) a 27. ~~( ) 2 A/Cl ( )9 2P-1) c2 o(-p)= a cot. T aa=l P He found for A = 4p, p= 4m +1, a prime > 1, (f2) V~pCz(-4p)-~ P-(yr 3 5. - (2) VC(-I ) sin v2r/(2p)- 2 ( 3.,-2) For A-=8p, Lerch derived more complicated formulas which are analogous to (1) and (2). L. Gegenbauer239 in a paper on determinants of m dimensions and order n, stated the following theorem. If for k= 1,..., n in turn in a non-vanishing determinant of even order m, we replace, in the sequence of elements which belong to any particular 236 C. F. Gauss, Disq. Arith., Art. 252; G. L. Dirichlet, Zahlentheorie, Supplement IV, ed. 4, 1894, 313-330. 237 Sitzungsber. Akad. Wiss. Wien, 104, IIa, 1895, 103-137. 238 Sitzungsber. Bohm. Gesells. Wiss., Prague, 1897, No. 43, 16 pp. 239 Denkschrift Akad. Wiss. Wien, Math.-Natur., 57, 1890, 735-52. 11 156 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI rth index, the elements which belong to the oth index /c, by the corresponding elements respectively which have the oth index 1c2 + k + A, where - A is a negative fundamental discriminant and where all the indices are taken modulo n; and if we divide each of the resulting determinants by the original, the product of VA bly the sum of the quotients has mean value, G(- -), when n becomes infinite (p. 749). Three similar theorems include a case of n finite. M. Lerch240 employed sin 2vX7 E*(x)== —4- - y=1 V7r Then E*(x)= [x] if x>0 is fractional, but =-[x]- if x is an integer. In the initial equation, x is replaced by x + am/A, where - A is a negative fundamental discriminant; each member is then multiplied by (- A/a) and summed for a= 1, 2, 3,..., A-1. Since [a misprint is corrected here], (1) (-) (A > ) - ( A a=w:( a ) -a a ) it follows from the theory of Gauss' sums (cf. G. L. Dirichlet, Zahlentheorie, Art. 116, ed. 4, 1894, p. 303) that ( A-i)E*(+ a)- J1(^)a(^)A ) (Y(^)-A2)c a=l\ a Hi *A A \ a / m / 7 v v Then by Kronecker'sl71 formula (5,) we have (2): l()E* + + ) mC(- ) - o V(V _). v a=i a V A T m /r v= i r By comparing this result with the case m = 1, we have for x= 0, 2( ) T - p m C/(-A)- - ( )E*(am) T M a=i a For m not divisible by A, E*(am/A) is equal to [am/A]. Taking m=2 and applying (1), we get241 2 2 -(4) [2-(^) (-A)= Hereafter we take A>4, i. e., T=2. Then, for m=4, we have ( [ (A )0 a=j a A [ \ When we put S(a,..., b) for: (-A/a), formula (5) is reduced by means of (1) to Buto..()Ais+,( equ nt, t 3S (0, 4 4 2 4-4 * 2 A) But (4) is equivalent to S(,..., A +S(4,.,-)=[2-(2 C(-A,). T 4 2 AL -\ Ir~ — l 240 Bull. des sc. math. (2), 21, I, 1897, 290-304. 241 Cf. H. Weber,220 Gottingen Nachr., 1893, 145. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 157 By combining the last two formulas we obtain the two serviceable ones A still more expeditious formula is obtained by taking m- 3 in (3), whence and this relation combined with (6) yields a=[I/4]+ / a (+ / 3 \( Cl For m = 1, (2) becomes ()T C( -1 v v 2 __ - \A Cos 2V x( /1 (8) -CM( -A)_ < A4 This is a generalization of Dirichlet's19 formula (1) and it holds for -A not a fundamental discriminant. Lerch showed that (8) is valid for any negative discriminant when 0 ~ x< //A by reducing it from Dirichlet's19 formula (1). By simply integrating (8), he deduced T( -/) - v) s 2 for ~F(4kc) and eF(4k-1), summed for c=l1, 2,..., n, where F(A) denotes the number of classes of discriminant -A. He identified these results with the concise ones of lermite211 which had been obtained from elliptic functions for forms ax2 + 2bxy + cy2. Lerch24 in an expository article, deduced for negative and positive discriminants Dirichlet'sh class-number formulas (1)i in which enters P(D) = (D/h)/h. For an arbitrary discriminant D, where D\ =A, he found by logarithmic differentiation of the ordinary r-function that P (D) -- / r k M. ~erh24 a~plid t Brneeer orm As by 2 h n ustuio d 242 Rozpravy ceske Akad., Prague, 7, 1898, No. 4, 16 pp. (Bohemian). 243 Rozpravy ceske Akad., Prague, 7, 1898, No. 5, 51 pp. (Bohemian); resume in French, Bull. de l'Acad. des Sc. Boheme, 5, 1898, 33-36. 158 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI To this he applied the identity: T(\//\ r./) Ar -i a1 2a/on. car r( )/ r( -) r '(1) -_log2A- - cot + k cos -- l ogsin. a=l For the fundamental discriminant Do, this furnishes familiar formulas including, e. g., for D> 0, Weber's220 formula (1). Lerch244 repeated the deduction of his240 formula (8) and established the validity of the formula for a non-fundamental discriminant D for the interval 0 ~ x< 1/(zAoQ'), where D'=-Ao.Q and Q' is the product of the distinct factors of Q. Lerch245 transformed the Gauss sum n-i N e2a2mir/l, a=0 as it occurs in class-number formulas (cf. G. L. Dirichlet, Zahlentheorie, Arts. 103, 115) and so obtained finally ~(1) {~ a2m -E am n- ( 2 Cl(-d) =1 n n \d T where m, n are relatively prime positive integers, n is uneven and q2 its greatest square divisor, while d ranges over the divisors of n which are -3 (mod 4). Lerch has since274 repeated the deduction in detail. From (1) follows 74 (2): -1+ am E (+ )} - si — (-D-1)- m 2 Cfl(- 4di,) -(/3Cl(-d3), 24 d_, \ I 7d, d3\ Td/3 in which d1 and d3 range over the divisors of n such that d = 1, d3 3 (mod 4). J. de Seguier246 in a paper primarily on certain infinite series and on genera simplified his results by substituting the class-number for its known value. He found, for example (p. 114), if F(x) is an arbitrary function which insures convergence, then 1 DoQ2 /D\ jz-j) n2\ ^ F (am2 + bmn + cn2) K (DOQ' ) a, b, c ( Q O(D, d)r(Dod2) (Did\) (-D -)F(d 12hL ) d K(D- d2) hk- hLit where K(m) is the number of properly primitive classes of discriminant m; A is representable by am2 + bmn + cn2;. D-=DD2 = DoQ2,. being fundamental; and 0(Di, d) is the number of classes of discriminant D, and of order d, where dd'=Q. *J. S. Aladow247 evaluated in four separate cases the number G of classes of odd binary quadratic forms of prime negative determinant -p: 244Rozpravy ceske Akad., Prague, 7, 1898, No. 6; French resume in Bull. de l'Acad. des Sc. Boheme, 5 1898, 36-37. 245 Rozpravy 5eske Akad., Prague, 7, 1898 No. 7 (Bohemian). French resume in Bull de 1'Acad. des Sc. Boheme, Prague, 5, 1898, 37-38. 246 Jour. de Math. (5), 5, 1899, 55-115. 247 St. Petersburg Math. Gesells., 1899, 103-5 (Russian). CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 159 (i) If p=7 (mod 8), G equals the difference between the number of quadratic residues and non-residues ~< p-3-2 (3/p) \. (ii) If p 3 (mod 8), G equals the difference between the number of quadratic residues in the sequence ~(p+l), 4(p+5),..., 12p-3-(3/p) and the number in the sequence ^p+3-2(3/p),.., i(p-3). (iii) If p 5 (mod 8), G equals twice the difference between the number of quadratic residues and non-residues in the sequence Xp-+3+2(3/p) A, ~1p+9+2(3/p) a..., I(p —l). (iv) If p =1 (mod 8), G equals twice the sum of the difference between the number of quadratic residues and non-residues in the sequence t(p+3),..., 32p-3+ (3/p) and the corresponding difference in the sequence J p+3+2(3/p) a, fp+9+2(3/p),..., 1(p-1). R. Dedekind,248 in a long investigation of ideals in a real cubic field, proved the following result. If at least one of the integers a, b, ab is divisible by no square, and if we write kc=3ab or k=-ab, according as a2- b2 is not or is divisible by 9, then the number of all non-equivalent, positive, primitive forms Ax+ Bxyy + Cy2 of discriminant D-B2 -4AC= -3c2 is a multiple 3K of 3. For primes p-1 (mod B), p not dividing D, K of the forms represent all and only such primes p of which ab2 is a cubic residue, while the remaining 2K forms represent all and only such primes p of which ab2 is a cubic non-residue. D. N. Lehmer249 calls any point in the cartesian plane a totient point if its two co-ordinates are integers and relatively prime. He wrote r pi-l r P(m, k) = DE o(~_ n p_+_ i p. i==1/^ 14 —l(23m~l>' iv ' - — z=1 The number of totient points250 in the ellipse ax2 +2bxy+cy2= N, b2-4ac= D= -A, is (1 ) V aP(, 2A); 7r and in the hyperbolic sector, always taken251 in this connection, the number is (2) i2.VDP(l, 2 D)N log(T+ UVD), N being very great in both cases. Noting now Dirichlet's93 formula (2) for the 248 Jour. fiir Math., 121, 1900, 95. 249 Amer. Jour. Math., 22, 1900, 293-335. Cf. Lehmer,218 Ch. V, Vol. I. of this History. 250 Cf. G. L. Dirichlet,20 Zahlentheorie, Art. 95. 251 Cf. ibid.,19 Art. 98, ed. 4, 1894, 246. 160 HISTORY OF THE THEORY OF NUMBERS. rCHAP. VI number of representations of a given number by a system of quadratic forms of determinant D, he finds the class-number, for example, for D -A <0, -r 1 1 N h(D)=E - V-P(1,2A)lim V 1'()8(x)-ooin which E is the number of solutions of t2-Du2=1; x is any positive number relatively prime to 2D, v(x) is the number of distinct prime factors of x; ~ (x) =1 or 0, according as each prime divisor does or does not have D as a quadratic residue. K. Petr,252 by the use of five functions A (=Hermite's69 A'), B, C, D, E, all analogous to Hermite's69 X deduced all of Kronecker's54 eight classic relations. For example, from expansions by C. Jordan (Cours d'analyse, II, 1894, 409-411), he obtained (1) ~031@2()1(v)( = C-.1()-8 cos(2n+l)- v ('q)7 i -kq-. 2 (v)1 1 Also C is the coefficient253 of 2q~ cos 7rV in the product of the right member of (2) - (v) (v) =21 sin 2n7rvq2'.- 2q-+2q-f... +.2q-( by the right member of254 (3) 61~3 @(v) 4q sin 'rv 4q3/2 sin 3rv 4q5/2 in 5Tr (3) 0103 _qi + 07rV O) y (v) = 1-q 1- q3 1-q5 But in that product, the coefficient of cos vry is a power series in q in which the coefficient of qN+~ is 8 times the combined number of solutions of n2_ (ko+~)2+ (n-1) (21+1) =N+i, 2- (k+I )2+ (n+l) (I +1 -N+t where n and I are positive integers, I taking also the value zero; k =0, 1, 2,..., n-1. But these equations can be written255 in the forms (4) { (r —k+1)(ni+7) + (n-k-1) (1+1) + (n —k) (+-1)=N, ( (n-c) (n+k+1) + (n- k) (1) + (+ k+ 1) (1) =N; and the left members may be regarded as the discriminants N=ab+bbc+ca of reduced Selling255a quadratic forms a(y-t)2+b(t-x)2+c(x-y)2, in which a, b, c do not agree in parity. Since there is a correspondence between such Selling forms of discriminant N and odd classes of Gauss forms of determinant -N, we have (5) C= 8F-(n)qn. The identity (Fundamenta Nova, ~ 41) 0)___2 _2,) - nq nq cos 27rnv 2 @3~l(v)~> 1 -- q12' - q^ 252 Rozpravy ceske Akad., Prague, 9, 1900, No. 38 (Bohemian); Abstract,261 Bull. Internat. de l'Acad. des Sc. de Boheme, Prague, 7, 1903, 180-187 (German). 253 Cf. P. Appell, Annales de lEcole Norm. Sup. (3), 1, 1884, 135; 2, 1885, 9. 254 Cf. C. G. J. Jacobi, Fundamenta Nova 1829, p. 101, (19); Werke I, 1881, 157. 255 Cf. J. Liouville,88 Jour. de Math. (2), 7, 1862, 44; Bell,370 and Mordell.372 255a E. Selling, Jour. fur Math., 77, 1874, 143. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 161 is multiplied member by member with Jacobi's expansion formula for ~, (v). In the resulting left member, the coefficient of cos Trv is C. 2q1~3. When this coefficient is equated to the coefficient of cos rrv in the resulting right member, a comparison with (5) yields the relation: (I) F(n) +2F(n-12) +2F(n —22) +... =dx -Sd, where dx denotes a divisor of n which has an odd conjugate and d, denotes a divisor of n which is < Vn and which agrees with its conjugate in parity. He also found the classic formula54 for the number of solutions of x2+ y2 +z2=n. To obtain a class-number relation of Liouville's256 second type, Petr expands in powers of v each member of an identity of the same general type as (1) above. Coefficients of v2 are equated, with the result that 12F(8n -12) + 32F(8 —_ 32) + 52F(8- 52) +... =2nAdx- 2n(, (di i+ x ) -:S(d + ax), where, the d's are the divisors of 2n; d < V/2n; di is odd; dx has an odd conjugate; and the subscripts of d retain their significance when they are compounded. To obtain a class-number relation of Liouville's257 first type, each member of an identity of the same general type as (1) above is expanded in the neighborhood of v=1. Equating coefficients of v, Petr then obtains H(8n ) - ( 3) +(8n — 152) -... =: (- 1)(d'+d'2+1)d,, where dl is a divisor of 2n such that its conjugate d' is of different parity, and d' < V2n. K. Petr,258 employing the same notation as252 in 1900, multiplied member by member the identity 02 ~ (v) = 8( (-1)" (n + Ic) qn2-i+(2n+l)sin (2k - 1)rv, =0, 1, 2, 3,...; c=1, 2, 3,.... by the formula for transformation of order 2 0 (v)~ (v)/e~2(0, 2r) =0(2v, 2T). In the resulting left member, the coefficient of q- cos 7rV is 16QF(n) q'"2 (0, 2r); in the right member it is 8 times the sum of (_1)n+-l (n+2k) q(n+2k)-2k (-1)+- (n(+2+ - )q("+2k-1)2-2(k-l) for n=0, 1, 2, 3,...; k=1, 2, 3,.... Hence (1) - 1) vF (n-2 V2) =-:(- 1) x+-1x, where x and y are the integer solution of x2 - 2y2 = n, x 2y, y 0 0; while, as also in 256 J. Liouville,l07 Jour. de Math., (2) 12, 1867, 99. Cf. G. Humbert,293 Jour. de Math., (6), 3, 1907, 369-373, formulas (40)-(44), as numbered in the original memoir. 257 Cf. J. Liouville,107 Jour. de Math. (2), 14, 1866, 1; also G. Humbert,293 ibid. (6), 3, 1907, 366-369, formulas (35), (36). 258 Rozpravy cske Akad., Prague, 10, 1901, No. 40 (Bohemian). Abstract, Bull. Internat. de 1' Acad. des Sc. de Boheme Prague, 7, 1903 180-187, (German). 162 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI (2), v ranges over all integers, positive, negative, or zero. In the summation x receives an extra coefficient 1 if one of the inequalities becomes an equality. Similarly, (2) Y(-1)vF(8n —82) =(-1))(+v)y, X2- 2y2=8-1, x>2y, y>O. These are the first published class-number relations which are obtained from elliptic function theory and which involve an indefinite quadratic form, e. g., x2- 2y2. By means of the elementary relation 7~010203 =27r =(-1)(2n+1)q ()2, n=O, 1n, 12, 3,... and the relation o~3 = 4PF (4n + 2 ) q4(4n+2) the identity ~(3.2 —@~O@203. ~0 yields F(4n+2) -2F(4n+2-4.12) +2F(4n+2-4.22) +... = (- 1)'(X-12, x, y>O, x2+y2 =4n +2; which is of the type of Hurwitz.202 A transformation formula of order 3 in a treatment similar to the above yields five such relations as F(4n+3) -H(4n +3) -2[F(4n+ 3-3.12) -H(4n+3-3. 12)] + *..=(-1)(x+8y-1)y, x2+3y2=4n+3, x O, y>O; and (3) F(4n) - 2F(4n - 3 12) +2F(4n-3 -22) -.. = -22x, 2 -3y = 2n, y > 0, x ~ 3y. From transformations of order 5, Petr obtained three relations including, (4) F(8n) -2F(8n- 512) +2F(8s-5n22)-.... -4 -x 2- 5y2= 2n, y > 0, 5y C x. M. Lerch259 wrote - o 1 R(o, s) = (+ V)', K(a, b, c; s) -'(al2+ bnn +cn72)-s, v==0 (o ) where o is an arbitrary constant; m, n=0, ~1, ~2,..., except m=_ =0; (a, b, c), a positive form of negative discriminant -A; a, b, c real. From Dirichlet's20 fundamental equation (2), it follows that the relation (1) a K(a, b, c; s) =A-sR(l, s) Y( R, s) a, b, ~ r=l r is valid over the complex s-plane, if (a, b, c) ranges over a system of representative primitive positive forms of discriminant -A, which is now supposed to be fundamental. 259 Comptes Rendus, Paris, 135, 1902, 1314-1315. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 163 Employ the Maclaurin developments in powers of s, (2) R(w,s)=( -)+log (s+...* 2 '+"7. (3) Kl(a, b, c; s) = 12s log H ()ff ( )] + where 00 H(o) = ewin12 (1 - e2nwi). 1 When substitution is made of (2) and (3) in (1), Lerch compares the terms which are independent of s and obtains Kronecker'sl71 class-number formula (5). E. Laudau260 showed that every negative determinant <-7 has more than one properly primitive reduced form (cf. the conjecture of Gauss,4 Disq. Arith., Art. 303) by proving that if — =b - ac is <-7, there is always another such form in addition to (1, 0, A). If there is no properly primitive reduced form (a, 0, c) other than (1, 0, A), then A has no distinct factors, but must be of the form pX, p a prime. (I) If p=2, and A > 4, there is the additional properly primitive reduced form (4, 2, X-2+1). (II) If p is an odd prime and if there is no reduced properly primitive form with b 1, then A+l cannot be expressed as arc, where one of the factors is uneven and >2. Hence A+-=2". When v 6, there is an additional properly primitive reduced form (8, 3, 2v-3 + 1). Landau now tested the few remaining admissible A's and found none which are > 7 and have a single class. K. Petr261 gave in German an abstract of his two long Bohemian papers,252, 258 including eleven class-number relations of the second paper. He indicated completely a method of expanding ~~'1 (0, 5S), which leads to new expressions340 for the number of solutions of x2+y2 +z2 + 52 =n and hence to generalizations of Petr's258 relation (4). M. Lerch,262 in order to find the negative discriminants -A for which Cl ( -A) = 1, wrote - A= AoQ2, where Ao is fundamental and q ranges over the distinct factors of Q =Q'llq. Then th on the equation to be satisfied is (Kronecker,171 (4)) Cl(-A) = 2 Q { q-( )}ol(-A^) = To q q If Ao=4, then T=4, Q=1 or 2. If Ao0=3, then To=6, Q=1, 2 or 3. If A0>4, then To=2. Here Cl(-A) can be uneven only for Q'=1 and Ao prime, or for A = 8. The case Ao=8 is excluded if Q =v 1. If Ao is a prime, C (-A) >1 unless Q=q=2, (2/Ao) =1, i. e., Ao= 8k-1. But if 7c 2, (1, 1, c) and (2, 1, c) are non-equivalent reduced forms of discriminant - Ao. 260 Math. Annalen, 56, 1902, 671-676. 261Bull. Internat. de I'Acad. des Sc. de Boheme, Prague, 7, 1903, 180-187. 262 Math. Annalen, 57, 1903, 569-570. 164 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Hence Cl(-A) = 1 for. =4, 8; 3, 12, 27; 8, 7, 28. Any further solution A must be a prime 3 (mod 8). But it is undecided whether there are such solutions other than 11, 19, 43, 67, 163. Lerch263 wrote +(x) for r'(x)/r(x) and observed that Dirichlet's23 formula (7) for the number of positive classes of a positive fundamental discriminant D gives the relation D(D)^ = - - VD Cl (D) log E(D). From this 1 is eliminated by means of -C-log 47+log a-q(x) -f/(1- x) o f XM 2dz 00 dz = J a-(-+m)2Z +2 cos 2nx7r e-nz" -, m= -oo Jl/a Vz n=l a where C is the Euler constant330 and a an arbitrary positive constant. The final result is that C1 (D) is determined uniquely by S-2(Pr+Qr) 2< () < S- 2(Pr,+Qr) log E(D) log E(D) in which, to a close approximation, S=1/VD(log D+.046181) — log D+.023090, 2 2 9 V _ ~t ~ 2 2edx Q dx ~P,= ^ r 7e-X 2dx, r 2 Ir e-/ D s<r / 3\/7/D SVD/r 2 r '/D X while r is chosen sufficiently large to insure a unique determination of C0(D). For example, if D=9817, logE(D) =222, S=450.5, whence CI(D) <450/222. We need not compute Pr and Qr since CI(D) is uneven (Dirichlet93) and hence is 1. J. W. L. Glaisher264 called a number s a positive, a negative or a non-prime with respect to a given number P, according as the Jacobi-Legendre symbol (s/P) = + 1, -1, or 0. He denoted by ar, br, Ar, respectively, the number of positives, negatives and non-primes in the r-th octant of P. For example, if P= 8k1 +1 is without a square factor, Dirichlet's23 formulas (5) for the number of properly primitive classes of determinant -P and -2P, respectively, h'=2(a,-b, +a2-b2), h"=2(a,-b —a4+b4) become265 h'=4(a, +a2) -(P-l), h"=4 (al-a2), where ar =ar+ IXr. Similarly for other types of P. Obvious congruencial properties (mod 8) of h' and h" are deduced from all of these formulas. Again h' and h" are expressed in terms of fl=bb+-2Xr (r=l, 2, 3, 4). Next, ir and ur are used to denote respectively the number of positives and non-primes <P 263Jour. de Math. (5), 9, 1903, 377-401; Prace mat. fiz. Warsaw, 15, 1904, 91-113 (Polish). 264 Quar. Jour. Math., 34, 1903, 1-27. 265 Glaisher, Quar. Jour. Math., 34, 1903, 178-204. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 165 which are of the form 8k +r, while Lr =r+ I-r. A table (p. 13) transforms the preceding formulas into results such as P=8+ 1, h'= 2 (L1-L3), h" = 4(L1-L7). If Qr denotes the number of uneven positives in the rth quadrant plus 2Xr, we have, for example, P=8k+l, h'=2(Q4-Q2), h" =4(Q4- Q1). L. C. Karpinski266 gave details of R. Dedekind's267 brief proofs of his theorems which state the distribution of quadratic residues of a positive uneven number P in octants and 12th intervals of P in terms of the class-number of - P, - 2P and - 3P. He added to Dedekind's notation the symbols C5 and C,, which denote the number of properly primitive classes of determinant -5P and -6P, respectively, and, by an argument precisely parallel to that of Dedekind, obtained for all positive uneven numbers P which have no square divisor, the distribution of quadratic residues in the 24th intervals of P as linear functions of C, C2, C3, C,. He put St=S(Sr/P), where t is a positive integer, and sr ranges over the integers x for which (r- 1)P/t<x<rP/t. He deduced such relations as the following: If P= 23 (mod 24), (1) S=-S6 =C,, S6=S6-=S6S6=0. If P 1, 5 or 17 (mod 24), C3 is a multiple of 6. For P= 3 (mod 4), 1C S0, + S10 + S,10 C + S19~ - 25. c1s sy1y-sy 0 C0 =2S S +4 40+2S10. G sl~+s}~+ '~ ~o, _ - 2 3! Cf. Dirichlet,23 (5). Three other relations among S~0 which arise from familiar properties of quadratic residues lead to a complete determination of S}~ as linear functions of C, and C, for r1=, 2, 3,...,10. E. Landau268 studied the identity D 1 r(1- )27r D\ 1 Y, — 1 /VAoCOSn n-I' D<0 n=1 n n1bs 7r 2 —t n1 which is valid for a real s, 0<s<1. The limit of the right member for s=0 is 7 -1\n/ nl The customary evaluation of the divergent left member for s=0 would give (Dirichlet,20 (1) above) the erroneous result h=:S (D/n). A similar study is made of the limit for s = 0 of the ratio D log n (D\ whic fo i enl n=which f213 limit ratio. n which for s= 1 is Kroneeker's213 limit ratio. 266 Thesis, Strassburg, 1903, 21 pp.; reprinted, Jour. fur Math., 127, 1904, 1-19. 267Werke of Gauss, II, 1863, 301-3; Maser's German translation of Disq. Arith., 1889, Remarks by Dedekind, 693-695. 26 Jour. fur Math., 125, 1903, 130-132, 161-182. 166 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI *M. Lerch269 denoted by g an arbitrary primitive root of a prime p== 2m + 1, and put p=l F,(x) =: a ind t xv =l1 where a is an integer of index m =p-1-n referred to the primitive root g as base. C. G. J. Jacobi270 had found the relation Fm(l+y) — Y/m! (mod p), where Ym is the sum of the terms in y"i, ym-l,..., yl-1 in the Maclaurin's expansion of [log(1+y)]n. Thus P-l/ V1 1v)v — y 1-l) (mod p) Hence if cj is the coefficient of yJ in Y (y), and if we set 12m -. cv(i-1)"=A+iB, M! v=m then A- B- H (mod p), in which H is the number of positive quadratic forms of discriminant - 4p. H. Poincare271 wrote F(q)= S qam2+2bmn+cn2 q e-t m, n where (a, b, c) is a fixed representative properly primitive form of negative determinant - p and the summation is taken over every pair of integers m, n, for which the value of (a, b, c) is prime to 2p except m=n-=O. F(q) is regarded as a special case of the Abelian function (A ) ~(x, y) =:ei (m<+nv) q"Im+2bm'n+cn2' The theory of the flow of heat is used to show that if k, c' each range over all integral values, ~(x, y) may be written (B) ~ (x), )e -- b 2- =-p, k, k' Et P=-Et [a(y/-27Cr)2-_2b(y-2_k'r) (x-2k7r) +c(x-27 2r)2]. Now for x=y=O and t small, ~(x, y) is asymptotically F(1). Hence, in the neighborhood of q= l, 2w F(q) 1t and is therefore independent of the choice of (a, b, c) of determinant -p. But, for p a prime =-3 (mod 4), we have (cf. Dirichlet's20 formula (2)) a(2) - - qPS(2b) )qnn', (a,b, c) m, n \ P 269 Bull. Int. de l'Acad. des Sc. de Cracovie, 1904, 57-70 (French). 270 Monatsber. Akad. Wiss. Berlin, 1837, 127; Jour. fur Math., 30, 1846, 166; Werke, Berlin, VI, 1891, 254-258. 271 Jour. fur Math., 129, 1905, 120-129. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 167 where now P=am2+2bmn+cn2 ranges over a system of properly primitive forms of determinant -p, and m, n take all pairs of integral and zero values for which P is prime to 2p; in the second member, n, n' range over every pair of odd positive integers each prime to P. By a simple transformation of each member, (2) can be written (3) S F(q)- S F(-)-=4s(IL (a, b, c) (a, b c) q2n 1 But from (A), it follows that F(q) = (0, O), F(-q)=-~(7r, 7r); and hence from (B), it follows that F(q) Fe-q P(- 9)=st e- P 72a/2cV2), ) -Et ) t E2t (a -2blv+cv'), where a, v are even integers in the case of F(q), and odd integers in the case of F(-q). Since for t small, all terms of the left member of (3) except those having = v 0 are to be neglected, the left member becomes _ p h(-p). tVp Moreover lim tq~ t=o 1-q2 - 2n Hence272 (3) becomes Dirichlet's14 formula (2). Equation (3) is also transformed to give Dirichlet's23 closed form (5) for h(-p). A. Hurwitz273 by the substitution aX + P/y + 71Z a,x + 32y + 72z ax +/3y - yZ ' ax + /y +y -Z transformed the Cartesian area l du dv of a plane region G into what he called the generalized area of G with respect to the form ax + fy + yz. Such a generalized area of the conic xy-2 = O is (1) 27r/(V/4ay- _ 2) 3. For points on the conic, we put x=r2, y=rs, z=s2, and consider points (x, y, ) = (r, s) = (-r, -s), r and s being relatively prime integers. An elementary triangle is one having as its three vertices the points (2) (r, s), (ri, si), (r+ri, s, ), rsi-ris= +l. All such possible triangles in the aggregate cover the conic simply six times and their total area is (3) 1 ar2+ rs+ys) (ar+3rs +Iys2) [a(r+r) 2+ (r+rl) (s+sj) +y(s+S~)2] -I summed for the solutions r, s, rl, s, of rs, -r1s = 1. 272 Cf. G. L. Dirichlet, Zahlentheorie, Art. 97. 273 Jour. fir Math., 129, 1905, 187-213. 168 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI But if the Gauss form au2 +,uv + yv2 be subjected to all the unitary substitutions, it goes over into aWu2 + 'u 'v' + y'v'2, where a', /3', y' have values such that (3) can be written as S8/-a'ty'(at'+,'+-y') a, where (a', /'/2, y') ranges over all forms equivalent to (a, 13/2, y). Hence by comparison of (1) and (3) we have 37r_ h= - 1 2DVD a,b,c ac(a+2b +c) ' where (a, b, c) ranges over all positive forms of determinant D. By modifying his definition of generalized area Hurwitz obtained for the right member a more rapidly convergent series. M. Lerch,274 by use of his240 trigonometric formula for E*, showed by means of Gauss sums that n-l 2m n1/ a2Ma2M n S- m E* (x+a )= + (x+ )_ n +S a=00=0 a=\ 2 S, in which S1 is the imaginary part of S,=x -(1 \)/ dr h/ d-' ^ i-I(d' —1)2e2dV^'xri, d=l VV ' dv I where dp is the g.c.d. of n and v, and dl= n/dr, v'=v/dy. Then, if we put d=d d, = d', and also I(z,d)= dv-K cos)2vzr, if d —1 (mod 4); =V d -)sin — -- if d- +1 (mod4); y=1 V'7 we find. /= (d-)(dx d). dim d Hence we get the chief formula of this memoir: X 2M M?)-)} + a MY n (=1)' X x n )n 2 d But by Kronecker,171 (2), i(O0, A) =2rT-1C(-A), where CI(-A) denotes the number of primitive positive classes of discriminant -A. And for x=0, m positive and relatively prime to n, (1) becomes275 Lerch's formula245 (1). For x=-, (1) becomes276 ranging over the divisors 4 3 of. Similar results are obtained by taking d ranging over the divisors 4 3 of n. imilar results are obtained by takingx= x= (cf. Lerch245 (2)) and x=1. 274 Annali di Mat. (3), 11, 1905, 79-91. 275 Cf. Lerch, Rozpravy ceske Akad., Prague, 7, 1898, No. 7; also Bull. de l'Acad. des Sc. Boheme, Prague, 1898, 6 pp. 276 Reproduced by Lerch in his Prize Essay,278 Acta Math., 30, 1906, 242, formula (40). CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 169 Lerch observed that the sum A of the quadratic residues of an odd number n, which are prime to n, >0 and <in, is given by 2 -[1+[ )1 l[ +1l(.. [l+ ( K^P)] ViL \ piJ L V/ where p2, P2,. P are the distinct prime divisors of n. Hence d v=1ld )()\v where d ranges over those divisors of n which have no square factor. By means of the Moebius function (this History, Vol. I, Ch. XIX), he transformed this into 2wA =-1n~(n)-n -Cl(-d)Md(n), d 'd where d ranges over those divisors -1 (mod 4) of n which have no square factor and ld (n) = Tl- (p/d) a, where p ranges over the distinct prime factors of d'= n/d. M. Lerch277 in a prize essay wrote an expository introduction on class-number from the later view-point of L. Kronecker171; and stated without proof that if.S (x)=x- Ix] and g(m, l) = S Pi () p=1 M and if - A1 and - A are two negative fundamental discriminants, and D= AA2; moreover, if for an arbitrary positive integer r, t and u be defined by (T+ U~/D _ t+u\VD then 2T(U A bu-t t au 2 TIT.,2= v v a, b, (a,b, c)[a (u 2 f 12a 4 ] where (a, b, c) ranges over a complete system of representative forms of discriminant D, a>O. In Ch. I, use is made of Dirichlet's20 fundamental formula (2) to make rigorous Hermite's83 deduction of Dirichlet's23 classic class-number formula (5). By new methods he obtained the familiar evaluations of the class-number that are due to Dirichlet,23 Kronecker,171 Lebesgue,36 and Cauchy,29 and established anew Kronecker's171 ratio (4) of CI(D,.Q2) to Cl(Do). He found that if Di are fundamental discriminants (z=1, 2, 3,..., r), and ID I = Ai, and if 2v of the determinants are negative, then (1) CT (DD2... Dr) log E (DD2..Dr), 1.....\, (Dhl. ahi h, h+r) 277 Full notes of the Essay were published in Acta Math., 29, 1905, 334-424; 30, 1906, 203-293; Mem. sav. etr., Paris, 1906, 244 pp. HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI where O<hi<Ai, the term containing log is to be suppressed, and E(D))= (T+ UV/). By taking r=2, D= -A, D2= -4, we obtain one of the corollaries: (AQ-1) )/ - tan hr/a Cl (4A) log E (4A) = 2S ) log ltan h/A h=1 tan hA In Ch. II, Lerch extended his240 methods of 1897 and obtained new formulas including the following comprehensive formula, suitable for computation: [DI2] D) [aA/D] A) _ a —1 a v=1 v where -A, D are fundamental discriminants, and A, D> O. Also, C2 _(- _ )=_ _0 _ r(_ 2 ( ) r V= V )V r(, O<x<l. In Ch.278 III, the identity in cyclotomic theory279 A (x) _ 0/ D\ X log V — DsgnD S (-), Hx|<1, B (x) 0 ^==i\ p. I /A where D>0 is a fundamental discriminant, for the limiting value x=1, gives, by Kronecker,l71 (2) above, the formula, a (1) + A DZ (1) 2 log IY (1 + 1 Z ( 1) I cl(D)log E(D)-=log (1) DZ() log. Y(1)-VDZ(1) V4F(I) Suppose D is prime and >3; if in the known identity 2 (1) -DZ2 (1) =F (1) = 4D, we put Y(1) =-Dz, Z(1) = y, we get y2-Dz2 =-4; and hence y and z do not satisfy the equation t2 - Du2 =4. Hence -YI + 12 logT+ U log ~ V 2log T~ is not an integer. Therefore C0(D) is odd. Similarly, it is proved that if D is >8 and composite, C1 (D) is even (cf. G. L. Dirichlet,93 Zahlentheorie, near the end of each of the articles 108, 109, 110). Congruences (mod 2) are given for C (- 8mn), m a prime. Lerch showed (Acta Math., pp. 231-233) how to obtain Y(x, D1D2) and Z(x, D1D2) from the cyclotomic polynomials for Di and D2, and thence found for D1, D2 fundamental and > 0, (DD2) E (DD_) = ( ) log Y(q, D1) + VDtZ (g, D1) _ i/2 DD=1)o hg 1 (\, O,)- VDZ(q, D,) Lerch obtains the following as a new type of formula analogous to Gauss sums: (1-l _ 1 (1) E cot -7 =4m -CZ(-), y=1 m 8 6Vo where -m is a negative, fundamental, odd discriminant, and 8 ranges over the divisors of m which have the form 47k+3 (Acta Math., 1906, 248). 278 Chapters III, IV appear in Acta Math., 30, 1906, 203-293. 279 Cf. G. L. Dirichlet, Zahlentheorie, Art. 105, for notation. CH:AP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 171 To express CTl(D),where D is fundamental, negative, and uneven (Acta Math., 1906, pp. 260-279), as the root of congruences (mod 4, 8, 16,...), Lerch put D=DDi2D3..Dm, where the Di are relatively prime discriminants, and put A = ID, At= I Di. All possible products D't= DDr,... Dr and their complementary products Q'=D,.Za+1D.+2...D,. are formed and A' is written for \D'I; also, we let F(D')=,-CI(D'), if D'<0; =0 if D'>0, (D", Q') = I[1- (D/q) ], q ranging over the distinct divisors of Q'; and (D', 1) =1. Then (2) 2 (A) - s —(D' Q')F(D ), where 2*s denotes the number of those of the integers s= 1, 2,..., A which satisfy (Di/s) = 1 for all Di simultaneously. For example, when m= 2, D= -p, D2 -+ q, p and q being primes, p= 3, q 1 (mod 4), then the last formula becomes 4P pq - -Cl(-pq)+ I- j - (-). pq 1 P TV7p Since 1(p-l) (q-1) is -0 (mod 4) and C(-p) 1 (mod 2), we have Cl(-pq) =l-(q/p) (mod 4). Lerch also obtained congruences for Cl(-pqr) modulis 8 and 16. In Ch. IV, a complicated Kronecker relation in exponentials applied to Lebesgue's36 class-number formula (1) gives finally the following result: czl(-A,).Cl(-A2) T71T2 m=l=l n=l n + a-~- >1 (m ) — 1n) in which s=- 2mnrri//(A2o), t=nu7ri/(A2w), while u, o are complex variables, the imaginary part of o is real and, in the complex plane, u is in the interior of the parallelogram with vertices at 0, 1, 1 +, o. Lerch specializes the formula in several ways. For example, for A, A, u =0, w=i, it becomes [Cl(-_A)]2= - (2 )A( e 2n/ 27r n=1 n where ~1 (kI) is the sum of the divisors of 1c. H. Holden,280 in the usual notations281 for the cyclotomic polynomial, wrote 4X =4XX =2 pZ2 H=h(-p)/[2- (2/p)]. 280 Messenger Math., 35 1906, 73-80 (first paper). 281 Gauss, Disq. Arith., Art. 357. 12 172 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI For p a prime of the form 4n+3>0, he found, putting h =h(-p), (1), (2) (x- sl (-(1) (-1)(- p d2 =- (-1) &-) 3)) (4) dx _7, (dx2I=.-1- 2, (5), (6) r 1 -iVpH, 1 1- =()iph, a, />0, <p, (a/p) =1, (/3lp) =-1, r=e2/ilp. The fifth formula had been obtained in a different way by V. Schemmel.282 The fifth and sixth are true also when p = 4n+ 3 is a product of distinct primes. Holden,283 by a study of the quadratic residues and non-residues, transformed the Schemmel-Holden formula (5) above into LI\' IJ /pq a 2p/q / 2-lp/Q\I (7) [q- g ) I=- (q-l) + (q-f-2) S +-..+ + (-) \pJ a=op/ P/gqP (q.-2)!p/q\ where q is any positive integer relatively prime to p; and the last series terminates with the last possible positive coefficient. If q=3 and q=4, (8) becomes [3- ()1 ] = 2 (- ), 3ff =3 () + P/ ) L P a= —0\ \P/ p/4 \P the latter284 being Dirichlet's23 formula (5i); for, the first or second term of the second member vanishes according as p = 8n +3 or 8n +7. When q =2, 3, 6 successively, (8) becomes three equations which yield H=21 (V)/{l+(-p) + ( )-(p)} a0 0 - - (r-1)p/\P6 p in terms of H. When q=2, 4, 8 successively, (8) leads to linear expressions for H in terms of the distribution of quadratic residues and non-residues in the first four of the octants of p. When q=p-1, (7) yields Dirichlet's23 formula (61). By taking q=2, 3, 4, 6, 12 in (8), a table is constructed which shows an upper bound for h when p 7, 11, 19 or 23 (mod 24), as h < (p+5)/12 if p 7. 282 Dissertation, Breslau, 1863, 15: Schemmel,95 (6). 288 Messenger Math., 35, 1906, 102-110 (second paper). 284 Cf. Zahlentheorie, Art. 106, ed. 4, 1894, 276. 285 Cf. Remarks by R. Dedekind 127 in Maser's German translation of Disq. Arith., 693-695. Cf. L. C. Karpinski,266 Jour. fur Math., 127, 1904, 1-19. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 1.73 Dirichlet's23 formula (61) is transformed into i (P-3) H=-(p-1)(p-2)-2 [V/kp], p=3 (mod 4). kc=l Holden286 multiplied each member of (7) by (p/q). The result for q=4 or q=2, p= 3 (mod 4), is reduced to ( = 3:), ( + ), [l- 2 ( -1 E =) 4n:( _<4n+ -) Hence h 9Z(c-l)o-<( -)(n _3 (mod 4), n odd, n<p. He found eight similar expressions for h including the cases of determinants -D, where D=4m+3, 2(4m+ 1), 2(4m+3) is a product of distinct primes. Holden287 for the case p=4n +3, a prime, put (' _Ji ) 1 + + 2m + + ( I ' 1+ -l+r l++rg r 4 + +. l g 1+r-2' where w is a primitive root of xP-l= 1, g is a primitive root of xP-1 1 (mod p), and r is a root of xV= 1. Then (6) becomes: ( i<^-l)? i^ ^ 1. A (2)t+/ A study of the new symbol gives A=1.i )P(p-l) [ -I- ]i()A,, where XA is the number of positive integral solutions k, 1 < - (p-1) for a given jt of the congruence kc +1 0 (mod p), and b is the number of quadratic non-residues <jp of p. Similarly, p-I h2=- E (s/1p),X. A=1 Holden288 in a treatment similar to his first paper280 obtained from his own transformation289 x- 1 ___= S2 ( _ -)(P)pxT2 of the cyclotomic polynomial, six expressions for h. For example, if p is prime, ( )I = ( - ^(-1)-(p-a)~ l- p}2 (~-.,)= - - ~, according as p 3 or p = 1 (mod 4). 286 Messenger Math., 35, 1906, 110-117 (third paper). 287 Messenger Math., (2), 36, 1907, 37-45. 288 Ibid., 36, 1907, 69-75 (fourth paper). 8s9 Quar. Jour. Math., 34, 1903, 235. HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Holden290 removed the restriction of his second paper283 that q be relatively prime to p. He put p=nP, q=nQ, where P and Q are relatively prime, and found that, if p= 4m +3 is free from square factors, then for any positive integer n, h 1+ ( a) =a+a3+a5+...,+ ~ 1 -- (- =a2+a4+a6+... where ar (O<r ~ n) is the sum of the quadratic characters of the integers between (r-1)p/(2n)+1 and rp/(2n). As above,283 he found that (p-3) is an upper bound of h for p= 3 or 15 (mod 24). Holden,291 by a modification of his second paper,283 obtained, when p=4n+ 1 is a product of distinct primes, Dirichlet's23 formula (5); also writing ar=- P (p n=(r-1)p/q with q prime to p, he found in the respective cases q=8, q=12, =(-p-)^2 It - 1[+(3)]h. In particular, q=8, p= 8n+ 1, h=-a-a3; p=8n+5, h=a2-a4: q-=12, p=24n+ 1, h=a,-a=-a-a2= -2(a3+2a4); p=24n+ 5, h=2a3=-2(a4+as); p =24n +13, h- = al = = -a= - a6; p=24n+17, h=2(a-a4)-=-2(a3+2a4). E. Meissner292 supplied the details of the arithmetical proof by Liouville90 of a class-number relation of the Kronecker type. G. Humbert,293 following Hermite,69 wrote o Y./:q1(4N+3)f(4N+3), 4N+3=- (2m+1) (2m+4p+3) -4p2, 0 /-=0, ~1, ~2,...; n, p=O, 1, 2,..., and recalled that the exponent of q has a chosen value as often as there are quadratic forms -= (2nm+ 1)x2+4 4uxy+ (2m+4p+3)y2 =ax2+2bxy+cy2 satisfying the conditions c>a, Ibl<a, a and c uneven, b even. By means of the modular division of the complex plane, he set up a (1, 1) correspondence between the principal roots of these forms and those of the reduced uneven forms of determinant -(4N+3). Hence f(4N+3) =F(4N+3). Similarly Humbert employed, and C to mean the same as -C and I+ D in the notations of Petr.252 290Messenger Math., 36, 1907, 75-77 (Addition to second paper283). 291 Messenger Math., 36, 1907, 126-134 (fifth paper). 292Vierteljahrs. Naturfors. Gesells. Zirich, 52, 1907, 208-216. 293Jour. de Math., (6), 3, 1907, 337-449. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 175 A new class-number relation analogous to Kronecker's54 (VIII) is deduced by equating coefficients of qN+j in the identity 2e-i7/8y (iV q) q.:q(8v+7) + F +7) 0 ==48 ^lom Q [qt(2m+l)2'( _ )n(rn-l) + qJ(2n-i)2(_ 1)(rn-l) (mr-2)]* o 1+q2m The result is ( $+1 F)[8N- (2x+l)2]= 8( 8)" where 2N= s8, S<8, 8 and 8, positive and of different parity. Similar treatment leads to relations of the Kronecker-Hurwitz type294 such as ( -1) I(n+1)F[8N+4- (2M+l)2] =( 1)(a-l)a, m>0 a ranging over the solutions of 2N + 1 - a2 + 2b2, a>0. Four class-number relations of Liouville's107 first type are obtained, including two of Petr,295 and also s8(-l)[ E (-)m(2 m+ 1-)F(4N+ 1-(2m + )2] =-2( )d2+(a2-4b2), in which 4N+ 1=a + 4b2; a>0; 4N+1=dd', d < d'; the term in which d=d' is divided by 2. New deductions of five of Petr's296 class-number relations of Liouville's297 second type are given (pp. 369-371). Like Petr,258 by recourse to transformations of order 2 of theta functions, but independently, Humbert obtained class-number relations involving the forms x2 2y2, including Hiumbert's (57), which is a slight modification of Petr's258 (1) above, and including Humbert's (52), which is Petr's258 (2) above. A geometric discussion, analogous to the one above in which Humbert evaluated <, now shows (pp. 385-8) that for a negative determinant -M, M =-3 (mod 8), there is a (3, 1) correspondence between the proper and improper reduced forms. The corresponding well-known relation (Dirichlet20) is similarly established for M=_7 (mod 8). To prove a theorem of Liouville,105 Humbert finds (pp. 391-2) in Liouville's notation that, for a determinant- (81M +3),,a(a'- a) =2(2mn m',+2mm' + 2mnm — m,-m -m, ), where a and a' a are the two odd minima of any odd class, while m,, m', ml' denote the first uneven minima of the three odd classes corresponding to a single even class, and where summation on the right is taken over the even classes. But the right summand equals 8M+3, whence Sa(a'-a) =-(8M + 3)(8M+ 3). 294 L. Kronecker,124 Monatsber. Akad. Wiss. Berlin, 1875, 230; A. Hurwitz,202 Jour. fur Math., 99, 1886, 167-168. 295 Cf.252 Rozpravy ceske Akad., Prague, 9, 1900, Mem. 38. In Humbert's memoir the two are (35), (36). 296 Rozpravy ceske Akad.,256 Prague, 9, 1900, Mem. 38. 297 Jour. de Math.,107 (2), 12, 1867, 99. The five are numbered (40)-(44) by Humbert. HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI To obtain class-number relations in terms of minima of classes, Humbert equated the coefficients of qN+, in the identity,/O =-qmnam[1-2q2m+....+(-l)pq2pm+..., 1 where am= q-1/4 _3q-9/4 +...+ (1)" n- (2 ) q-(2"-1)2/4, 0= (-1) q 2. -oo The coefficient in the first member is S (-1) +VF (4I + 3 - 2-4x2 y2). x, y In the second member, 4VN+3= 4m2- (j-l)2 +mp, (m 1, p>, 1 _ 1 m); and the coefficient is (-1) l +P-12(2u-1). When 4N+3= (2m+2p-2j+ 1) (2m+2p+2 -l) - 4p2 is identified with ac-b2 the negative of the discriminant of form (a, b, c); a and c uneven; c>a; a>b; b>O; the latter coefficient is 2n,(_l) (c-a+2b-2)1(c- )-2(-1)1V(_2 ) (1) nl) where the summation on the right is over the proper classes of determinant- (4N - 3), and 1,fL2(mX _ 2) are the two uneven minima of a class. Similarly, from i tOt Humbert obtained 4 (- 1 )F[ 4N-7 x2- (2y l)2] =2 ( - l) ( L+2f+2) summed over all pairs of integers x, y, where f.is the even minimum, /i,, /2 the odd minima of an odd class of determinant -4N. By equating the coefficients of qN in the identity m2q q2 m'+m. -4XL(- )e/ -Sm=8S _q2 +8~2m2 q 2 [1+2q-1+.. +2q-(-l)2] 1_ q2rnz 1- q2m we obtain the class-number relation (-1)+l_ (_l)7 (4N-42 -1) (L - 1)= V2, h>0 where b(n) is the sum of the divisors of n, and I is the even minimum of an odd class of determinant - 4N. Similarly, from the expansion of ('fiO, it is stated that 1[4N-1-(4h+3)g "'(ih+3) =-t t2 h>0 L 4 where I(D) =F(D)-3F,(D), and FP(D) denotes the number of even classes of determinant -D. Five more new class-number relations involving minima include 8 s F[8M+3-(4h+3)](4h+3) =r2v(v-vV), h>O in which t(n) denotes the sum of the divisors <Vn of n; the summation on the CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 177 right extends over the even classes of -(8M+3); and vi, v2, vs are the three minima of a class, vC v2: v3. To obtain class-number relations of grade 3 of the Gierster'45-Hurwitzl67, 184 type,298 Humbert employed the fundamental formula of Petr299 and Humbert,300 (1) rlOHH12/~2 -2,H1 (x, Vq)Sq(s8+7)/8F(8, + 7) 0 4q(2m+1)2,.[ (2 - 1) q-(2m-1)2 + (2m-5) (2-... ]cos(2m+ 1)x. 1 By setting x= 0, and equating coefficients of qN, we obtain (2) O=F0 =F[8N- (2m + 1)2] - 2 (8- ), 2N=881, 81 even, 8 odd, 8<81, m arbitrary. In (1), we put x;= r/3 and use the formula for ~(3x,q3). In the resulting identity we equate the coefficients of qN and use the fact that the number of solutions of 8N7= (2x+1)2+ (2y+1)2+3(2z+1)2+3(2t+1)2 is 16Sd', where d' ranges over the divisors of N which have uneven conjugates and which are not multiples of 3. Whence, for N = -1 (mod 3), the final result is -3Sd'= 2SF [8N- (2m + ) 2]cos(2m + 1) r/3 -4 (8-8)cos(8+ 8)7r/3 in which 2N = 88, so that cos (8 + 8)7r/3 =. This result combined with (2) gives, for N- -1 (mod 3), the relations30l (p. 418):.F[ASN-9(2/+ 1)2] -=d', F[8N - (6p+_1)2] =- -d'+ -2:(8a'-8), summed over all integers l/, p, where d' is a divisor of N which has an odd conjugate and 8'8= 2N, 8>8, 81 is even and 8 uneven. Corresponding results301 are obtained for N-0, 1 (mod 3). Transformations of the third order yield also, for N=61 +1 (p. 431), 2a G (N -9v2) = r + 4d, summed over all integers v, and all divisors d of N, where G(m) is the number of classes of determinant -m, and A3, is the number of decompositions of N into the sum of 4 squares in which 3 of the squares are multiples of 3. Humbert evaluated such sums30' as F(N- 9v2), with N arbitrary; but it is done with less directness than by Petr.345 New expansions lead to such relations301 as (-1 )mF(24N-1-24m2) = — y(-1/y) (-1)~(-2), 48N-2=x2-6y2, y>O, x>3y; 6 ( _l )Dh[24AT+ 1-(6n + 1) 2 _ (3)( 3, ) 24N 1= x2- 6y2, y 0, x> 3y, each summed over all integers m. Terms in which y=0 are divided by 2. 298 Cf. Klein-Fricke,217 Elliptische Modulfunctionen, II, 1892, 231-234. 299 Cf. Petr,252 formula (1). 300 Numbered (10) in Humbert's memoir. 301 Humbert gave the results also in Comptes Rendus, Paris, 145, 1907, 5-10. 178 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Humbert302 gave five new class-number relations involving minima303 of the classes. H. Teege304 partly by induction concluded that, when P=8n+3 is a product of distinct primes, (P-3 )/4 a (P-D)/2 (P-1)/2 (/ \ (P-3)/4 a(a) 5 S - )a+ ~ (a-)a>0.16124PVP, -a- (- 0. 1 \/ (P+1)/4\ (p+i)/4 I These combined confirm, in view of Dirichlet's23 formula (6), Gauss' conjecture (Disq. Arith.,4 Art. 303) that the number of negative determinants which have a class number h is finite for every h. K. Petr305 recalled that the number of representations of any number N by the representatives (Dirichlet,93 (2)) of all the classes of positive forms ax2+bxy+cy2 of negative fundamental discriminant D) is r: (D, d), summed for the divisors d of N, where the symbol (D, d) is that of Weber306 for the generalized quadratic character of D. Hence,307 if D< -4, (1) S qa2+b Y+cY2=2 qN(4(D, d))+h, ql<l, class x, y N d x, y=O, +~1 +2,...; N=1, 2, 3,... where h is the number of positive classes of D. By methods of L. Kronecker308 he obtained 00 00 i 0 00 (2) ~ ~ eriT(am2+bmn+cn2) - - g-47r-Tl(aM2+bl+cn2)/D m=-coo n=_oo 7T ID _-oo -co where r= - 1/r. Next, by the use of theta functions, he found c=1 @(kT/D, T) N=1 d d is any divisor of N. Now (1), (2), (3) imply (4) (D 7c)3~'v(kr/D, r) =2=i " ~ q-2(ax2+bxY+C2Y)/D_ 2hjri (4) S (D, 3c) (krlD 7 el 2 Y ()(D,) (/D,T),____ 2V A-D s s q:q(aX2+by/+c'2) 2h,-ri, T Cl, y where T= -1/r and q, = er1. For the same transformation 1= — 1/r, (5) (D ) [2 kri+ '(kr/D, ) _ 1 (D ) '(/D, T,) -c=1, 2,..., - D-1. By use of (5), we get (6) 2i [ (D, )]-2 q( c) L k D 7 T Clx,y 1 7 /)'D ' ) /T ( O 71) 302 Comptes Rendus, Paris, 145, 1907, 654-658. 303 Jour. de Math., (6), 3, 1907, 393-410. 304 Mitt. Math. Gesell. Hamburg, 4, 1907, 304-314. 305 Sitzungsber. Bohmische Gesells. Wiss. (Math.-Natur.), Prague, 1907, No. 18, 8 pp. 306 Algebra, III, 1908, ~ 85. 307 Cf. H. Poincare, Jour. fur Math., 129, 1905, 126. sos Sitzungsber. Akad. Witss. Berlin, 1885, II, 761. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 179 Now the right member of (6) is the product of 1/r by a power series in q,. Hence the quantity in brackets in (6) is zero (Dirichlet,19 (1)). For, otherwise (6) would imply that r= - ri/log q, could be expressed as a power series in q, which converges for all q, such that Iq| <1. Moreover, the comparison now of the two members of' (6) in the light of (1) gives Lebesgue's36 formula (1): h =- (D, 7c)cot. An alternative form of (6) is the following: 2J- h + E L (D, k) }= II 2rh\/-D+7r (D k) cot7rk/Z). The last two class-number formulas above follow now elegantly when r is regarded as a variable occurring in an identity. H. Holden309 applied the method of his first paper280 to a product p=4n+ 3 of distinct primes, and stated the four possible results including (d ) - (-1)2 <p(p) —1H. He generalized the method of his fourth paper288 from primes p=4n + 3 and 4 + 1 to products of primes, and gave the four possible formulas including /dT\ 1 p=4n+l, d) = =-(- 1) h, where m is the number of integers between Ip and 1p, and prime to p. H. Weber310 in a revised edition of his book on Elliptic Functions modified his earlier discussion214 of class-number to apply to Kronecker forms,m71 in which the middle coefficient is indifferently even or uneven. He also (~ 85) replaced the Legendre-Jacobi-Kronecker symbol220 (D/n) by (D, n) which he redefined and gave details (~~96-100) of Dedekind'sl27' solution of the Gauss4 Problem. M. Plancherel311 extended certain researches of A. Hurwitz312 and M. Lerch313 by finding the residue of Cl(D) modulo 2m, where D=D1D2...D, and D, D1, D2,..., Dm are fundamental discriminants. He deduced Lerch's formula314 2<m A m (1) ~(Zq), —y 8 (D - D1....DrDr...D a=-1 r,...,ra 1a a (r dl' where A= IDI, hA= ADiI; (D, Q) =I(1- (D/q), q ranging over the different prime factors of Q, and (.D, 1)=1; P(D) -- C(D) if D is <0, P(D) = if D is >0; and S* denotes that those values s only are taken which satisfy (D s) = (D/s)=... =(D /s) = 1. 309 Messenger Math., 37, 1908, 13-16. 310 Lehrbuch der Algebra, Braunschweig, III, 1908, 413-427. 311 Thesis, Pavia, 1908, 94 pp. Revista di fisica, matematica, Pavia, 17, 1908, 265-280, 505-515, 585-596; 18, 1908, 77-93, 179-196, 243-257. 312 Acta Math.,235 19, 1895, 378-379. 313 Acta Math., 30, 1906, 260-279; Mem. presentes par divers savants a l'Academie des sc., 33, 1906, Chapter III of the Prize Essay.278 t4 Acta Math.,278 30, 1906, 261. Lerch,78 (2). 180 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI Hereafter Ai are assumed to be primes. Then 14 (A) -0 (mod 2m-a). But (Dr, Dr 2. Dra l. A+. A ) =O 0 (mod 2"-a). It follows that P(Dr, D... Dr) =O (mod 2a-1), P(D1D2...Dm) 0 (mod 2m-1). The latter for D<0 is the rule derived from genera (cf. C. F. Gauss, Disq. Arith., Arts. 252, 231; L. Kronecker, Monatsber. Akad. Wiss. Berlin, 1864, 297; reports of both in Ch. IV). Thus (1) implies?(D>... Dm) —+ 1 (A...Am) m-1 r+ 1. (Dr...Dra, Ar a+.. Ar )P(Dr,.. Dra) (mod 211). a=l r ~, ra For a negative determinant D= -P1P2.. *P2m+iqq2... qn, where p, q are primes >0 and -p = q 1 (mod 4), this leads to (2) Cl(-pp'2...p2m+lqq2... qn) 2m+1 - S ) ((-1)p p%..Prpr-2 Pr Pr+. *I -P.,,q2 *. * * qcn),l=l r '(Pr(... Pr) (mod 22m+l), where the symbol ( I ) is defined by the recurrence relation (DD2... Da IDa+...Dm) m-a-1 =::(D1.. DaDPa+1... Ia+, APa++l. Apm)(D.. Da IDPl. DPa+), /a=O p and by the formula (Da DfDy) = (DaD,, A^) (Da, A8) + (DaDy, A1) (Da, Dy) + (Da, A3AY). He disposed completely of the new special case m =5 by (2) as in the following particular example: mn=5, D =-pi, D P2, 3 =-p3, D4 = q, D5 q2, (Pl\ - (P2 _P _ P3 Pi /~l 2 = / (ql q) ( q 1 q2 (2 q2/ The result in this case is Cz(D) -2(1- e) (1-'2) [1- (-4/-)] +2(1-~1~2) [2(1i-12) + (i-~) (1- (-4/a) )] (mod 32), where rl=(P2/P3), - 2=(P3/Pl)1, 93-(Pl/P2), 1- (q/q2), 'o='-l+w2+v3 -For D=DoDlD.2.. Dm, IDol 8 or 4, he obtained analogues of (1) and finally congruences (mod 2m+1) for Cl(D'). He noted that C1(-4qiq2...q) -0 (mod 2m+1) if each qi - 1 (mod 8). G. Humbert315 obtained formulas which express new relations between the minima of odd classes of a negative determinant -n and those functions of the type ( (n), 315 Jour. de Math., (6), 4, 1908, 379-393. Abstract, Comptes Rendus, Paris, 146, 1908, 905-908. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 181 x(n) of the divisors of n which occur in the right member of Kronecker's54 classnumber relations. Thus were obtained alternate forms for the right members of old class-number relations. E. Chatelain316 obtained the ratio (see the Gauss4 Problem) between the number of properly primitive classes of forms of determinant p2D, p a prime, and the number of determinant D. As the representatives of the first, he chose the type (ap2, bp, c) with c prime to p; as representatives of the second, he chose the type (a',', ') with c' prime to p. Then between the h(p2 D) forms (a, b, c) and the h(D) forms (a', b' c') he set up a (7c, 1) correspondence by means of a relative equivalence given by the unit substitution (a ), = 0 (mod p). Similarly he found the ratio of the number of classes of the two primitive orders of a given determinant. His proofs are similar to those of Lipschitz.41 M. Lerch317 gave two deductions of A-il _i\\-l n2 7 a- S (-1)a/k=4(1.-2E)K2-K, kc=2 \ / a-=l where - A is a negative uneven fundamental discriminant, K = 2tr-lC(-A), e= (2/A). Here, if A= 3, K=. The second and more elementary deduction rests on Lerch's240 formula (3). He deduced several formulas which he had published earlier,318 including htan -(-1)- K, A 0O (mod 8). h=l\ 'h h T K. Petr319 reproduced his305 discussion of 1907; and, by equating coefficients of qn in the expansion of doubly periodic functions of the third kind, obtained Schemmel's95 formula (4); also the number /~ of primitive classes of the negative fundamental discriminant - D = DD2 for D2> 0 and = 4 +1: h =- E (D,, k1) (D2, 72) EklEk,2 k1, k2 where ci =-0 1, 2,..., IDij -1; i=1, 2, and where eklE2== 1 or 0 according as c1/D1 + k2/D2>0 or <0, and (D, 7c) is the Weber symbol.220 Similarly, for -D= -DD1D2D3, a negative fundamental discriminant, h= -S(D1, c1) (D2, k2) (D, 7i3) E(l + + where i=1, 2,..., ID|-; i=1, 2, 3; and E(-a) = -E(a)-1 if a>0. These two formulas are special cases of a formula of M. Lerch on p. 41 of his prize essay.277 See Acta Math., 29, 1905, p. 372, formula (16). Cf. Lerch,277 (1). J. V. Pexider320 for the case of a prime p=8p+ 3, wrote r and p respectively for a quadratic residue and non-residue of p, and combined the obvious identity (1) 2,r+Sp= =p((p - ) with Dirichlet's14 formula (3), viz., (2) Sp-Sr=Ap, 316 Thesis, University of Zurich, 1908. Published at Paris, 1908, 79 pp. 317 Rozpravy ceske Akad., Prague, 17, 1908, No. 6, 20 pp. (Bohemian). 318 Lerch, Acta Math., 30, 1906, 237, formulas (36)-(39). Chapter III of Prize Essay.278 319 Casopis, Prague, 37, 1908, 24-41 (Bohemian). 320 Archiv Math. Phys. (3), 14, 1908-9, 84-88. 182 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI where 3X is the number of properly primitive classes of determinant -p. The result is - p-1 2 2 p — 2 p p 2 According to M. A. Stern, if p is a prime 41 + 3, there exists an integer u such that (3) 2Sr-Sp=ap. From (1), (2), (3), we get 3XA= - (p —1) -2o. This result compared with Dirichlet'sl4 class-number formula (3) shows that u is the number of quadratic non-residues of p which are <1p. For a prime p=8u-r+, (2) holds provided now A=h(-p). Hence by (3), h(-p)= (p-3)-2K, where K denotes the number of the quadratic non-residues of p between 0 and ~p. Dirichlet's14 formula (3) combined with the last result shows that B=K<s+(p+1), A=R+~(p+1), where A and R are respectively the number of positive quadratic residues of p less than ~p and ~p, and B is the number of quadratic non-residues < p. A. Friedmann and J. Tamarkine,321 in a study of quadratic residues and Bermoullian numbers, replaced!b-Sa in Dirichlet's14 formula (3) so that for p a prime — 3 (mod 4), the latter becomes Cauchy's28 class-number congruence (1) in the form322 h(-P) P ^ -(-2)] (_)-(p+1). * B(p+1)/4 (modp). M. Lerch323 found that, for P a prime, I cot ( - 1 a P VP' where a ranges over all positive integers <P prime to P such that (a/P) =1. Cf. Stern.31) G. Humbert324 introduced a parameter a in the 0-function, and considered H(x+a) and ~(a). Then, by Hermite's69 method, he found that +00 S (-l)kcos 2ka S cos (m2-m,)a= (- 1))Nd cos da, kF=-_ Cl(4N+3.-4k2) d where m, and m2 are odd minima (mn, mn2) of a reduced form of negative determinant - (4N+3-4k12), and d is a divisor of 4N+3 not exceeding its square root. For a=0, this becomes Hermite's9 relation (5). For a= rr, it becomes (-1) -- C(4N+3 16k'2) 2 kc=-oo C(4N+3 —6f'2)) ~(m2-m1) where (2/d) is the Jacobi-Legendre symbol. If N is uneven, this is Kronecker's54 relation (VII). 321 Jour. fur Math., 135, 1908-9, 146-156. 322 Mem. Institut de France, 17, 1840, 445; Oeuvres (1), III, 172. 323 Encyclopedie des sc. math., 1910, tome 1, vol. 3, p. 300. 324 Comptes Rendus, Paris, 150, 1910, 431-433. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 183 P. Bachmann325 supplied the details of Liouville's90 arithmetical deduction of a class-number relation of the Kronecker type (cf. Meissner292). M. Lerch,326 by a study of Kronecker's171 generalized symbol (D/n), transformed the left member of Lerch's240 formula (3), for a negative fundamental discriminant -A, and m not divisible by A, and found that a=l(a )T - - 1- } (1) (M-1) {l( K)}], K=- Cl(-). Put 2h fr,(a /ha\ -i - {E (2A)2E(i^)} Then by formula240 (4), we have K(A-1)_ / A_ 1-6 fl - n E-2 (2 ___ A hia =,n 2 —2 (2-E)K1 K, ( n=[a=-l a L Since h(- A) = (2 - )K, we have, for A = 2n +1, ( — a -h) h2 a=l a Similarly for A = 4P, 2 (-_1) (~- )( a_),p- ^,.P 1() ia = -E (2P )-E( a=l P \a-P M,4P For a negative prime discriminant - A, A = 4 + 3, (1) implies: (A —1) (-A)-= (-1)a. L. E. Dickson,327 by a method similar to the method of Landau260 in the case of Gauss forms, showed that for P>28 no negative discriminant -P=0 (mod 4) could have a single primitive class. For P=3 (mod 4), P with distinct factors, there are obviously two or more reduced forms. Hence, if there is only the one reduced form [1, 1, (1 +P)], then P=p, where p is a prime -3 (mod 4) and E is odd. But for p>3 and e > 3, a second primitive reduced form is [ (p+ l), 1, (pe+l)/(p+1l)]. For P=3e, E > 5, a second primitive reduced form is (7, 3, 9) or [9, 3, 1(3e-2+1)]. Hence beyond 27 we need consider only primes P. We set Tj- [(2j+1)2+P] =To+j(j+ ). For any integer m and any Tj, there is some Tr, 0 < r (m - 1), such that T- T, (mod m). From this lemma and by indirect proof it is found328 that there is a single reduced form of discriminant -P if and only if To, TI, T2,..., Tg are all prime numbers, where 2g + 1 denotes the greatest odd integers ~ VP/3. 325 Niedere Zahlentheorie, Leipzig, II, 1910, 423-433. 326 Annaes scient. da Acad. Polyt., Porto, 6, 1911, 72-76. 327 Bull. Amer. Math. Soc., (2), 17, 1911, 534-537. 328 Cf. M. Lerch,262 Math. Annalen, 57, 1903, 570. HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI When P =7 (mod 8) and >7, To is even and >2, and hence composite. A detailed study of P=3 (mod 8)= 8k-5 shows that for all P>163 some Ti is composite except perhaps for k=3t and t =5+ 12 or 51+13. With this result and by a stencil device Dickson showed that no P under 1,500,000 except P=3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163 could have a single primitive class. M. Lerch329 obtained the chief results of Dirichlet by simple arithmetical methods and reproduced the deduction of several of his240' 277 own labor-saving formulas. E. Landau330 established Pfeiffer's'95 asymptotic expression for K(x) =-Sn HI n=l where Hf denotes the number of classes of forms ax2 + 2xy + yy2 of negative determinant - n. Let HnI be the number of non-equivalent reduced forms of determinant -n and with!f- =v. Then for a given n, in each reduced form y > a - 2v, and v V~n/3. Thus x VW/3 x yn/3 K(x)- E X HnP= s (Hfon+ s HnV) n=1l v=0 n=l V=l Vs/3 X x Vx/3 Y= H no+ E S H,=V- Hno+ S R(x, v). n=l y=1 n=3v2 n=1 v=l But Hn, is the number of solutions of ay=n-, y - a. That is, if T(n) is the number of divisors of n, H,,o=~T(n), if n is not a square; but H1,o-== T(n) +1}, if n is a square. Hence X x (1) S H~no=- 2 T(n) +I[x] =xlogx+(C —)x+O(Vx), n=l n=l where C is Euler's constant (=0.57721...) and 0 (k) is of the order11 of k. For a given v>0, Landau evaluated R (v,)=: ) IEVn n —3v2 by noting that R (x, v) is the number of solutions of ay yv2 + x, y a 2v, each solution being counted twice when y>a>2v. Hence R(x, v) is the number of lattice points in the finite area defined by these inequalities in the ay-plane, lattice points in the interior and on the hyperbolic arc exclusive of its extremities being counted twice. The resulting value of Vx;/3: R(x,v,) y=1 combined with (1) now gives iK(x)-= 9-X- +- +0(x log x). If C9corresponds to K, but refers to classes having a and y both even, the result obtained is (x)= -r x ( og). ~ ( )- _ (X +o( log X). 3s9 Casopis, Prague, 40, 1911, 425-446 (Bohemian). 330 Sitzungsber. Akad. Wiss. Wien (Math.-Phys.), 121, II, a, 1912, 2246-2283. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 185 Landau,331 by a study of the number of lattice points in a sphere, found that if C,, is the number of solutions of u2 + v2 + w2 =, x S Cn- 437r+ 0 (X+E), n=l where-17 only the order of the last term is indicated and e is a small arbitrary positive quantity. But by Kronecker54 (XI) above, if F(n) denotes the number of uneven classes of forms ax2 + 2bxy + cy2 of determinant - n, then C = 8F (n), if n- 3 (mod 8); Cn=12F(n), in all other cases except n-7 (mod 8). In u2+ v2 + W2 = 1 (mod 4) evidently u: v: w=l:: 1:, 0:, 0: 1: 1 (mod 2). Hence X SF (n)= -X3 +O (zx+), n=-1 (mod 4). This holds also for nr 2 (mod 4); but S F(n) = X~+0(x-+e), n_3 (mod 8). J. V. Uspensky,332 by means of lemmas of the types of Liouville's,90 gave a complete arithmetical demonstration of each of Kronecker's54 classic eight class-number relations. See Cresse.374 J. Chapelon333 obtained a new identity derived from transformation of the 5th order of elliptic functions and with it followed the procedure of Humbert.334 He added to Gierster'sl35 list of class-number relations of the 5th grade two new ones and gave relations also for 4F(4N-x2), x=-5 (mod 10);:F(4N-x2), x -+~1 (mod 10); and for F (N- 25x2) summed over all integers x, where N-= 5/AN'- 0 (mod 10), and N' is not divisible by 5. He gave335 24 class-number relations for 4F(N-x2) and HT(N-x2) which are characterized by various combinations of the congruences N — 2, ~4 (mod 10) with x=0, +1, ~2 (mod 5). These 24 relations include Gierster's relations of the 5th grade.134 The right hand members of Chapelon's classnumber relations in these two memoirs are all illustrated by the following example for N _2 (mod 10): F(N -_X2) = 3S6/-i ( - 1)+'d'+ 1 (-1) d,(d - d), x-+~1 (mod 5), where d' is any divisor of N and N=ddd with d >d (see Chapelon's thesis340). G. Humbert,336 after giving an account (Humbert185 of Ch. I) of his principal reduced forms of positive determinant D, proved that for D = 8M + 3 S(-1/,)f(a-lbl)=2Sef(2c+l ), |-=lbl-(a+c), 331 Gottingen Nachr., 1912, 764-769. 332 Math. Sbornik, Moscow, 29, 1913, 26-52 (Russian). 333 Comptes Rendus, Paris, 156, 1913, 675-677. 334 Jour. de Math., (6), 8, 1907, 431. 35 Ibid., 1661-1663. 336 Comptes Rendus, Paris, 157, 1913, 1361-1362. 186 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI where f(x) isn n arbitrary even function; the summation on the left extends over all principal reduced forms (a, b, c) of determinant D; and the summation on the right extends over all decompositions, 8M+3-= (2c+1)2L (2k'1+l)2+ (27"+1)2, 7c, k', c" each 0, of 8M+3 into the sum of three squares. When f(x) = 1 and we employ the known value (cf. Kronecker's54 formula (XI)) for the number of decompositions, we have (8M + 3) -: ( -1//). If f(x) = X2, we have 8(8 M+3)F(8M+3) = (-1//3) /(a+c). G. Rabinovitch336a proved that the class-number of the field defined by V -d, where d= 4m - 1, is unity if and only if x2-x + m (x =,..., m- 1) are all primes. Fewer conditions are given by T. Nagel.336b G. Humbert,337 by Iermite's method of equating coefficients in theta-function expansions, found that, for all the negative determinants - (8M + 4-4k2), in which M is fixed, the number of odd classes for which the even minimum is not a multiple of 8 is the sum of the divisors of 2M +1. Similarly for determinants - (8M-4k2), the number of these classes is (8 +8) the summation being extended over all the decompositions 2M=88,, 8 odd, 81 even, 8<8,. Also, by Hermite's method combined with the use of an even function (cf. Humbert336), he338 obtained the following formula for the number F of odd classes having the minimum and the sum of the two odd minima = 0 (mod p), p arbitrary: 1) l)F(4N + 3 - 4r2) = ( 1) 'd, r<,, >0, and for h arbitrary, r is =h (mod p), 4N+3= pdd,, with pd<d1, and d, 4h (mod p). *F. Levy339 discussed the determination of the number of classes of a negative determinant by means of elliptic functions. J. Chapelon340 gave an outline of the history of class-number relations of the general Kronecker54 type and listed Gierster's35 relations of the 5th grade. Examples will be given here merely to characterize each of the six exhaustive chapters of the thesis. Chapter I contains theorems on the divisors of a number. Let N= 2AN'- 5^N= 295^N'', N' and N" prime to 2 and 5 respectively; N- did, d, > d; d' any divisor ofN;and, let =-Sd'(d'/5), = (-1)'d'(d'/5), 9 = (di-d) (di +d/5). Also let N= da, d> VTN, a< V/N. Then '[( a) 5+ ( a-)2 [+5v(aN 336aJour. fur Math., 142, 1913, 153-164; abstr. in Proc. Fifth Internat. Congress Math., Cambridge, I, 1913, 418-421. 336b Abh. Math. Seminar Hamburgischen Universitat, 1, 1922, 140-150. 337 Comptes Rendus, Paris, 158, 1914, 297. 338 Ibid., 1841-1845. 339 Thesis, Zurich, published A. Kiindig, Geneva, 1914, 48 pp. 340 These, Sur les relations entre les nombres des classes de formes quadratiques binaires, Paris, 1914, 197 pp.; Jour. de l'Ecole Polytechnique, Paris, 19, 1915. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 187 Chapter II gives, in Hermite80-Humbert293 notation, lists of standard transformation formulas for the ~-function and expansions of ~-functions and 0-functions. Chapter III presents fundamental formulas for the transformation of the fifth order of ~-functions. In o(5u, 5-) =-Cl O(u~ri/5), i=0, 1, 2, C1 is found to be v5/Y where s= (q5) and rq =(q) =E+ (-1) mqil (6m+l)2,5. Chapter IV deals with the representation of a number by certain quaternary quadratic forms. In (p. 90) "1g l = 8: 1q -~ 1-q^ cos 2mx, i l q2,n 1q2 m put x=7r/5 and x= 27r/5 and subtract. Equating coefficients of qM, we get (1 ) 5- (l(2M) ] - 10 + 2 where, (N) and 3' (N) are respectively the number of decompositions (in which the order is regarded) 4N= (2x+1)2+ (2y+l)2+ (2+1)2 5 (2t+1)2, 4N= (2x+ 1)2+5(2y+1)2+5(2z+ 1)2+5(2t+1)2. From another expansion it is similarly found that ~~~- (2) -12at+t>=S (N"5/5) 2A+2^ (0,/5), (2) W3 + W1 = 5 v (N, 0 summed over all divisors 0' of N"'. Then by (1) and (2), (N(-) =[L + +5Y+1 ("//5) ] [5, + I]. Suppose that N is even. Since for a fixed value of x, F[4N-5(2x+~1)2] is the number of positive solutions t, tu, v of t2+u2 +-2 =4N-5(2x+ 1)2 (p. 118), (3) E F(4N-5(2x+ )2) = IfFW1 (N)=312 [I + 5+l (N"/5) ] (5 Xi1). a=O0 This is a special case of Liouville'sl07 (4). In terms of functions like,i and A3 above, Chapelon found in Ch. V expressions for 5F(8M)-4-2), F(82)' ((M 4 - 2) S X( 2), ) (Nx2), N-X2 J(N-_2), J(N- 2 25,~25 F 2 where x-=5-(r-+ or 10IT +, k- constant; E(N) =F(N)-H(N), J(N)=F(N)+ 3H(N). In Ch. VI, Chapelon found sets of relations equivalent to each one of Gierster'sl35 relations of grade 5; and added large sets of new relations, the sets being distinguished by the residue of N modulo 10. He (p. 171) proved Liouville"14 (1). H. N. Wright341 tabulated the reduced forms ax2+ 2bxy + cy2 of negative determinant -A=D for A=1 to 150 and 800 to 848. The values of b, c occur at the inter 84 University of California Publications, Berkeley, 1, 1914, No. 5, 97-114. 13 188 HISTORY OF THE THEORY OF NUNMBERS. [CHAP. VI section of the columns giving a and the row giving A. For a given a, the reduced form occurs in periods, each period covering a values of A; and each period having the same sequence of b's. For a given D, the a's are found among those for which there is a solution of x2 -D (mod a). For the case of A without square divisors, he wrote r it V A=nk, a = hi 7^ki, 0 0 1 where h and kc are primes; ho = 2, i> 1 and the Vk's are those odd ki's which in a have exponents >1. Let v be the number of distinct factors kc' of a; let X be the greatest value of v for any a. Then for the given D, the number of reduced forms with a V/A-is found to be s f- {l- [ VA [ [V2+^)e ( Pe/) V=O iPo P IPo where 1i) is the ith product formed by taking v factors ck'; Po is a positive odd integer, Pe a positive even integer, both VA/; (D/Po) is a modified Jacobi symbol and if Pe = P'2, P' odd, then (D/Pe) = (D/P') (D/2), where (D/2r) is defined so that 1+ (D/2r) is the number of solutions of x2 _ D (mod 2r). The few remaining possible values of a which are > VA and V/4A/3 or [ 1[- +2V/ 1 +3A], according as a is even or odd, are to be tested by the most elementary methods. Examples show the advantage of this whole process over the classic one of Dirichlet,23 (5). E. Landau342 investigated the asymptotic sum of Dirichlet's series19 (ax2 +2bxy + cy2)-, in the neighborhood of s=1 for a form of positive determinant D. (For D<0, see Ch. de la Vallee Poussin, Annales Soc. Sc. Brussells, 20, 1895-6, 372-4). L. J. Mordell343 announced the equivalent of two serviceable identities of Petr344 in theta-functions. For, Mordell's Q and R are respectively Petr's C and 4.D. By specializing the arguments in the identities and equating coefficients of like powers of q, Mordell found new representatives of five types of class-number relations such as Petr252, 258 and Humbert293 had deduced. K. Petr345 combined C. Biehler's346 generalized Hermetian theta-function expansions, which Petr had used twice252', 258 before, now with W. Goring's347 formulas given by the transformation of the third order of the theta-functions. He obtained six expansions similar to the following348: 1 oo m-rB3 _ 0302) - -smi-/3((A ) +B()3()-_4 2cqn2 2 cos2m7r/3, 2(() sin2.7r/3 r3=2k=1 in which q=e ir, and B is found69 to be 82 qNF(N), where as usual F(n) is the 342 Jahresbericht d. Deutschen Math.-Vereinigung, 24, 1915, 250-278. 33 Messenger Math., 45, 1915, 76-80. 34 Rozpravy 6eske Akad., Prague, 9, 1900, No. 38 (Petr 252). 345 Memorial Volume for the 70th birthday of Court Councilor Dr. K. Vrby, 1915; Rozpravy ceske Acad., Prague, 24, 1915, No. 22, 10 pp. 346 Thesis, Paris, 1879. 847 Math. Annalen, 7, 1874, 311-386. 348 Cf. G. Humbert,293 Jour. de Math., (6), 3, 1907, 348. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 189 number of odd Gauss forms of determinant -n. On expanding 03(1-), it is found that the coefficient of qN in B3 is -4(2dx -Sd,) + 2S(d, -d)-6 vS(d (0)- d()) +12[F(N) +2F(N- 9.12) + 2F(N-9. 22) +...], summed for the divisors d of N; the subscript X on d denotes that the conjugate divisor is odd; the subscript 1 denotes that the divisor agrees in parity with the conjugate and is ~ VN; but, if it = VN, it is replaced in the sum by 1VIN. Also, N=d d2; N=d~ d~, d(~)+d'(2-O0 (mod 3). This includes the case N odd and =1 (mod 3) which G. Humbert349 had failed to provide for in a direct way. Similarly in B3 1 2 3)3 the coefficient of qN/9 is 8Y,(N- (9k-i)2 )- 23 d(6) -d(6)) where the subscript 1 has the same meaning as before, d) -d ) (mod 3) and where i=l, 2, 4, according as N 1, 4, 7 (mod 9). Alternative expansions of B3 and BB were obtained by Petr with indication of a method of determining in them the coefficients of qN and qN/9 respectively in terms of divisors of N and the number of integer solutions of x22 + y2 +2 + 9u2 =N and X2 + 9y2 + 9z2 + 9U2 =N, respectively. The class-number relations thus resulting were given by Petr in the next paper. K. Petr350 completed345 the deduction of the following class-number relations. For N arbitrary, F(N) +2F(N- 9.12) + 2F(N- 9.22) +... '=- 1 [dx + -d (~)-4Sd(o) + ( - 1) ( —1)+(Y:+l)x], summed over the positive odd numbers x, y satisfying 32 +y2 -4N, such that y is not divisible by 3. [Petr in this and all the following formulas of the paper erroneously imposed the latter condition also on x.] The upper index (0) indicates that the sum of the corresponding divisor and its conjugate is 0 (mod 3). Again for N arbitrary, F(N) -3H(N) +2[F(N-9.12) -3H(N- 9-12)] +2[F(N-9.22) -3H(N-9.22)]... = -(d —de) -~(dJ(~) - ~d()) + 2d0a)+ ~ (- 1)1(x-i)+i(+i)x where d agrees in parity with its conjugate divisor of N, and de is odd. For N 1, 4 or 7 (mod 9), the two following relations are given: FN- (9k-+-a) ) 9 7) didl (-1 )-(/-)+/() E (N-(9k~ta)) (N- (9c~+a))2] = -T (d- de) +- d1 + -~( - 1)(-1)+i(yl)x, in which a=1, 2 or 4 according as N-1, 4 or 7' (mod 9). 349 Jour. de Math., (6), 3, 1907, 431. 350 Rozpravy ceske Acad., Prague, 25, 1916, No. 23, 7 pp. 190 HISTORY OF THE THEORY OF NUMBERS. rCHAP. VI In equating coefficients of qN in the identity345 B3, Petr on his page 2 of the present paper employed the identity ~04(9)= 18~xq*(3X2+y2) ( 2~4(~) = 18:xq~(3+y2) ( - 1)(-) + (Y+-1) and failed to observe that x may be =0 (mod 3). So he introduced an error in the denotation of all the resulting class-number relations of the paper. L. J. Mordell351 deduced arithmetically the first class-number relation of his preceding paper343 in the form (1) F(m) -2F(m-12) +2F(m-22) -... (-1)i(ad)+ld, where d is a divisor < Vm of nm and of the same parity as its conjugate divisor a; but when d-= Vm, the coefficient d is replaced by Ad. Mordell considered the number of representations of an arbitrary positive integer m by the two forms (2) s2+n2+n(2t+ ) -r2=m, (3) d(d+28)=m, n>O, -(n —l) < rcn, t>O, d>O, 8 0. Then, if f(x) is an arbitrary even function of x, (4),( — l)rf(r +) = - 2(- l)df (d) where the summation on the left extends over all solutions of (2), and the summation on the right extends over all solutions of (3); but, when 8=0, the coefficient 2 is replaced by unity. Now take f(x) =-(-1)w. Then (4) becomes (-1) 8= -2 (-1) +dd. But for a given s, Mordell352 found that the number of solutions of (2) is 2F (m-s2). Hence we get at once the above class-number relation (1). Mordell352 illustrated his343 method by writing qo qn2~2n7rix f(x) = s -- + n= —oo ~^[ and proving that /(1) - \ (O ) - q)2+n (1- 2q-1 +. +..+2q-( -+-)q_ = (_)rq-r2+1(2t+l) n=(1) 21 —q2 n= ' where r —0, 1, ~2,..., ~ (n-1), n; t=0, 1, 2,.... But corresponding to each set of values n, r, t, there is a reduced quadratic form353: nxs2+2rxy+ (n+2t+ 1)y2 of determinant, say, -M. Conversely to each reduced form (a, 0, a) of determinant -M, there corresponds one solution, and to every other reduced form of determinant -M, there correspond two solutions, of the equation M=n2 —r2 + n(2t +1). Hence the right member of (1) is 2:; (-1)M F(M)q. When f'(0) is given its true value, and q is replaced by -q, and o01 by 1-2q+2q4- 2q +..., the 351 Messenger Math., 45, 1916, 177-180. See a similar arithmetical deduction by Liouville.90 352 Messenger Math., 46, 1916, 113-128. 358 And so this expansion (1) suggested to Mordell his351 arithmetical deduction. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 191 equating of coefficients of qn yields his351 relation (1); which is equivalent to Kronecker's54 (III), (VI), and is identically Petr's252 relation (II). Replacing f(x) by +(x) =f (x) Ogh(X+)/,oo(X), where ~ is an arbitrary constant, Mordell obtained the equivalent of Kronecker's (I), (II), (V). By the use of X(x) =f(x) Oo(2x, 2o))/Ooo(x), he obtained a class-number relation involving an indefinite form354 in the equation x2- y2= m. By the use of f(x)0oo (3x, 3) /0oo (x), he found (cf. Petr's258 formula (3) above) that F(2m) -2F(2m-3 3.12) + F(2m - 3.22) -... ( - 1) +1 X2- 3y2 ==, X>0, - (x-l ) y_ Replacing f(x), as initially used, by oo F(x)= S qtn2en7ri/(l1qn), n odd, n=-co he obtained 3[G(m) +2G(m- 12) +2G(m-22) +...]= -6a+4b+2 (-1)c, where a denotes a divisor of m which is ~ Vm and agrees with its conjugate in parity, but if a-= Vm it is replaced by a/2; b denotes a divisor of m whose conjugate is odd, and c a divisor of m whose conjugate is even. Kronecker's54 (IV) is the special case of this formula for m odd. G. Humbert,355 in a principal reduced form (Humbertl85' 186 of Ch. I), (a, b, c) of positive determinant with b>0, put 3=b-jla+cl, and, by Hermite's method of equating coefficients in O-function expansions,69 found that (51 (l) =2F(4n+2), 2 -1) =2F(8n+5), ( 2F(8n+l), where S1 extends over all the principal reduced forms of determinant 4n + 2 with a n and c odd; 22 extends over all the principal reduced forms of determinant n +5 with a and c even; S extends over all the principal reduced forms of determinant 8n+1 n with i(a+c) even. From the first of the three formulas is deduced the following: Among the principal reduced forms (a, b, c) of positive determinant 4n+ 2, the number of those in which b-l a+cl is of the form 4k+1 diminished by the number of those in which it is of the form 4k- 1 is double the number of positive classes of determinant - (4n+ 2). By denoting by Hl(n) the left member of the first of these three formulas, for example, and summing as to the argument 4M+2-(2s)2, Kronecker's classic formulas54 give H1(4M+2) +211 (4M+ 2-22) + 2H (4M- +2-42) +... =2, (4M+2), where Oi (n) is the sum of the odd divisors of n. 354 Cf. K. Petr, Rozpravy ceske Akad., Prag, 10, 1901, No. 40, formula (1) of the report 258; also G. Humbert,293 Jour. de Math., (6), 3, 1907, 381, formula (57). 355 Comptes Rendus, Paris, 165, 1917, 321-327. 192 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI L. J. Mordell356 recalled Dirichlet's20 formula (2). Whence357 if Irl <1, / F)\,r.k (1) S E r'iaw+bx =cA+r T- E; ()l-r ' a, b,c x,y kc=l 1-r summed for all pairs of integers x, y>O, =0 or <0, and for representative forms of negative discriminant D; while (D/7c) is the generalized symbol of Kronecker171 and A is the number of classes of discriminant D. We set r= e2Siw and write (1) as (2) q(o)-=A+x(o). When X(o) is evaluated in terms of 0-functions, (2) becomes:.1, (3) (O)=A+ 4T _((o) + V)} = (D) where 0(v) =-O,,(v). Nowa35 Hence when o is replaced by -l/, (2) gives Kronecker'sl71 formula (5i) for the class-number. E. Landau358s wrote e for the fundamental unit ~(T+ D/U) and by means of Kronecker'sl71 class-number formula (3), obtained an upper bound of log c//D log D for very great D by noting that K (D) - 1 and finding an upper bound of the sum of the Dirichlet series in that formula. E. Landau359 wrote h (lc) for the number of classes of ideals of the imaginary field defined by V -k. Let 8 be any positive number. If there are infinitely many negative values - (v) of -c<(k(/') <Ck2 <...) such that h(k) <k<-8, then, for every real o>l, k(v+1)>k() for every v exceeding a value depending on 8 and a. Given any c>l, if we can assign c, depending on o, such that, h (k) <cV<k7/log k holds for an infinitude of negative values - kI(" of - c, then kIc(^+ > 'k(v) for every v > 1. Known facts are proved about limits to h (c). He360 derived inequalities relating to h (k). G. Humbert361 let mi and m, be the odd minima of an odd Gaussian form (a, b, c), and H(M) be the number of odd reduced forms of determinant -M for which mi 356 Messenger Math., 47, 1918, 138-142. 357 Obtained independently by Petr,305 (1). 358 Cf. Mordell, Quar. Jour. Math., 46, 1915, 105. 358a Gottingen Nachr., 1918, 86-7. 359 Gottingen Nachr., 1918, 277-284, 285-295 (95-97). 360 Math. Annalen, 79, 1919, 388-401. 361 Unpublished letter to E. T. Bell, October 15, 1919. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 193 or m2 is 0 (mod p), p being a given odd prime; and if simultaneously m1-m- 0 (mod p), he let the class count 2 units in H(M); then, when N 0 (mod p), s(- 1)ITs(N-na2)) +=(- )nH(N-n2) = -2 (d/p) (-1l)( '+), where the first summation extends over all integers n 0 (mod p), the second over the positive integers n not =0 (mod p), and the third over all decompositions N=dd', with d-= 0 (mod p), d<d' and d, d' of the same parity. The class a(x2+y2), when a 0 (mod p), counts here as one unit in H(a2). Let oh (N) be the number of classes of positive odd Gaussian forms of determinant -N, for which the minimum ju is ~ 2h; if (L=2k, the class counts for 1 in fh(N). Then for N odd, positive, and prime to 3, 4o(N) +261i(N+3 12) +... ++h(N+3.h2) +... - 3 3 1 d where in the second member, the summations extend over all divisors d of N. In the first member, p7 certainly equals zero when h is > i (N+ 1). Similarly, N being odd, let,' (N) be the number of classes of positive even forms for which the minimum [t is 2/h; if t==2h, the class counts l in 0 (N). Then we have 'o(N) + 2c'( N+.12) ++212 +2 (N+ 2.22) +... +2h'(N+2h72) +... 6 2 d )+-4 (d) 3 () where, in the second member, the summations extend over all divisors d of N; (6/d) =0 if d-O (mod 3), and N' is the quotient of N by the highest power of 3 that divides N. And similarly,362 let ^h(M) be the number of reduced odd Gaussian forms (a, b, c) of determinant -M for which simultaneously a 2h, a+ c- Ib l 5h; if in these relations, there is a single equality sign, the form counts - in ih; if there are two equality signs, the form counts i. Then, if N 7, 17, 23, or 33 (mod 40), 0o(N) +2 (1V+ 5.12) +... +2 ^(N+ 5.h2) +... =-(-~/iN)Sd(-5/d), the summation extending over all divisors d of N. Class-number relations occur incidentally in Humbert's papers 18, 23, 24 of Ch. XV. L. L. Mordell363 deduced his364 formula (1) from the identity oO ezrt Ot2-2rtx1 \ 0=(ll, (,) _ -^r_ dt=f - - + (f (, -), where the path of integration may be a straight line parallel to the real axis and below it a distance less than unity, and where +-if (_1 ) (n-l) qtn2en e f()-n odd n odd 1 +- qn 362 Deduced by Humbert from his own formula (7), Comptes Rendus, Paris, 169, 1919, 410. 363 Messenger Math., 49, 1919, 65-72. 194 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI By applying Kronecker's54 formula (XI) to the right member of formula364 (1) and integrating the left member, Mordell obtained the relation364 (3). But by applying the identity oo (0, - 1/) = V -i 000 (o, W) to the right member, he found i 1 oo_ F(M) 2 + (n+a = 27 [4F(M)-3( G (M) ]e2- a r (a+M. 'a (~n~a) ~ M=O a1 L. J. Mordell364 announced without proof the formulas:!fc teriwt oo 2 ~ (1) J"e2st 1 dt= -2 F(n) qn+ -iuo)F(n) q +10 0(0, o ), - & v -1 1 2" fm te ta 2 X (2 ) -~ dte>+ l t 1)= >(-l ) nF (4n -) qi<'ff 'r~ _ V1 o _ 1 q) (2) oet~1 = (-+)(4n-1)V4"-i + - iV (-1)"-'F(n)q~, -o 2t? p L ~1 1 where R (i0) < O, q =e7iw, q= e-i/. Proofs were given elsewhere.363 By integrating, he deduced from (1) the relation, (3),4P (M)- 3G(M) 1 + 21 -4_ 2 F (M) e-2arVM + 2a: - (3) 2a2 1 (n+a) 1- o (a2f+M)2 where R(a) >0, a arbitrary. E. T. Bell365 proved that (1) m=4c-+l, N3(m)-=6[e(m) +42d~(m,-_2?) odd <Vm (2) m=4ck+3, N3(m)=8S2} i(m-v_2) X where N3(m) is the number of representations of n as the sum of 3 squares; e(n) =1 or 0, according as n is or not a square; and (n) is the excess of the number of divisors 4k+1 of n over the number of divisors 4k+3. He366 then stated that elementary considerations yield (3) m= 47+1, N3()=(m) = 6[() + 21$(m-4a-2)], (4) m odd, N3(2m) = 12 [ (n) +2(m- 2a2)], (5) m odd, N3(2m) =n12 (2rm-2 ), (6) n arbitrary, N3(n) =2[e(n) +28(n) +4(;4(n-a2)], where m, n, a are positive integers, K is any positive odd integer, and where x is as always >0 in E:(x). A comparison of (1), (2), (3), (4), (5), (6), with the well-known relations (Kronecker,54 (XI); Hermite,69 (7)) (7) Frm=4k+l, N3(mn)=12F(m); N3(n)=12[2F(n)-G(n)], ) m=8c+3, N,(m)= 8F(m); N3(2m) =12F(2m), 364 Quar. Jour. Math., 48, 1920, 329-334. 365 Quar. Jour. Math., 49, 1920, 45-51. 366 Quar. Jour. Math., 49, 1920, 46-49. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 195 where G(n) denotes the total number of classes and F(n) the number of uneven classes of determinant - n, gives immediately m=4k+l, 2G(m) =((m) +2S6(m-4a2), m= 4+ 1, 2G(m) =6e(m) + 4( r-(m-_2) A, m odd, G(2m) =-(m) +24S(m-2a2), m odd, G(2m) =S(2m-jU2), m=8k+3, F(m) =!S 2(m~-2), 12F(n)- 6G(n) ==e(n) +26(n) +4(4(n-a2). Similarly by comparing (7) with seven recursion formulas367 such as m =4k+l, N3(m) = 6 1(+(m+- 1) S- N(m-St), in which g (n) denotes the sum of all the divisors of n, and t>0 an arbitrary triangular number, he obtained the seven following recursion formulas for classnumber: m =4k+1, 2G(m) +2SG((m -8t) = i(,+1), m=4ck+1, 2G(m) +42G(m-4a'2) =l(m), m odd, 4G(2m)+4 G(2m-8t) =g(2m 1), m odd, G(2m) +2SG(2m-4a2) =4 1(m), m=8/c+3, F(m) +SF(m -8t) 4=i(m+l), m=8kc+3, 4F(m) +8SF(m -4a2) =g,(m), 6E(n) + 12EE(n- 4a2) =4(-1)"Al(n)- (n), in which X1(n)= [2(-1)n+1]l (n), where g'(n) is the sum of the odd divisors of n; and in which E (n) = 2F(n) -G (n). The last of these relations is equivalent to Kronecker's54 formula (X). L. J. Mordell,368 starting from Dirichlet's20 formula (1) ~h(-n)=Vn S -( ) r odd ' r and allowing for the improper classes, proved that Ij() - [\! -i( a/+bi) - (0( ) )j where the real part of io is <0; the radical is taken with positive real part; the summation is carried out first for a= 0, ~2, ~+4,..., and then for b-=, 3, 5,..., in this order; (a/b) is the Legendre symbol; but if a=O0, b=1, we replace (a/b) by 1. Also d is any even integer, c any odd integer, satisfying ad-bc1. He also proved that F(M)/VfM=f f(1) -if (3) +if(5)-..., where f(n) denotes the number of solutions of 2-M (mod n). Formulas of the same type are also given in which F(n) is replaced by G(n). E. T. Bell,369 by equating like powers q in the expansions of functions of elliptic theta constants, showed that the class-number relations of Kronecker, Hermite and 367 Bell, Amer. Jour. Math., 42, 1820, 185-187. 368 Messenger Math., 50, 1920, 113-128. 69 Annals of Math., 23, 1921, 56-67; abstract in Bull. Amer. Math. Soc., 27, 1921, 151. 196 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI others may be reversed so as to give the class-number of a negative determinant explicitly in terms of the total number of representations of certain integers each as a sum of squares or triangular numbers. Bell,370 by paraphrasing identities between doubly periodic functions of the first and third kinds, obtained three class-number relations involving a wholly arbitrary even function f(u)=f(-u). Let e(n) = or 0 according as n is or is not the square of an integer; let F(n) and F, (n) denote the number of odd and even classes respectively for the determinant - n, n 0. The first and simplest of the three similar relations is,(oa ) (f V( - /(d)- a' +d")] +25' a/ ( 2+ -d") - (d 8 + d")l =F(-4r2)f(2r)-( ) (_ _ +_d), the X,:' extending over all indicated positive integers a',..., 8 such that, for / fixed,, - 3 (mod 4), = a' + 2m" - d'A'+ 2d"8" (a'=d'a', m"=d"8"), and /=d8, d<-/p; a'=1 (mod 4), d'<V'; /-4r2>0. Interpreting results obtained by putting f(x)=O, xI >0, f(0)= 1 in the three relations, it follows that the total number of representations of any prime p by xy +yz- +zx, with x, y, z all >0, is 3 [G(p) -1] where G(n) =F(n) +FT (n); that the like is true only when p is prime; that there are more quadratic residues than nonresidues of the prime p-3 (mod 4) in the series 1, 2,..., ~(p-l); and so for p-1 (mod 4) inthe series 1, 2,..., (p-l). If f(x) =1 for all values of x, the first relation gives Hermite's69 (3): F(/ - 4r2) =- i(/8), where Is, (n) is the sum of the sth powers of all the divisors > /n of n diminished by the sum of the sth powers of all the divisors < V/n of n. For f (x) =x2, the first relation gives: 32sr2F(#-4r2) =- 3(#) + #i (#) - 321V(4l), the 5 extending over all integers r such that P-4r2>0, and N(4/3) is the number of representations of 4/3 in the form + m2 4+ Ml + M2 + 2m 2 + 2m + 2m2 + m2l for which the mn(i=l, 2,..., 8) are odd and == 0, and precisely 0, 2 or 4 of mi, m2, m3, m4 in each representation are included among the forms 8k ~ 1. The paper contains a table of the value of F(n), n=l,..., 100. E. T. Bell371 obtained 18 class-number relations which are similar to his370 three above and which form a complete set in the sense that no more results of the same general sort are explicit in the analysis. By specializing the arbitrary even functions which occur in these formulas, he stated that all the class-number relations of 870 Tohoku Math. Jour., 19, 1921, 105-116. 371 Quar. Jour. Math., 1923(?); abstract in Bull. Amer. Math. Soc., 27, 1921, 152. CHAP. VI] BINARY QUADRATIC FORM CLASS NUMBER. 197 Kronecker and Hermite and certain of those of Liouville and Humbert are obtained as special cases. L. J. Mordell372 showed that the number of solutions in positive integers of yz+zx+xy=u is 3G(n). It is shown essentially by Hermite's69 classical method that x+yl1 (mod 2) for 2F(n) of the solutions; x+y-2 (mod 4) for F(n) of the solutions; and x+y-O (mod 4) for 3G(n) -3F(n) of the solutions, where always a solution is counted I if one of the unknowns is 0. In particular, if n is not a perfect square, x+y l (mod 4) for F(n) of the solutions, x+y 3 (mod 4) for F(n) of the solutions. Particular cases had been given by Liouville88 and Bell.370 G. H. Cresse374 reproduced J. V. Uspensky's332 arithmetical deduction of Kronecker's54 class-number relations I, II, V and supplied some details of the proof. R. Fricke375 (p. 134) obtained and (p. 148) translated373 a result of Dedekindl27r in ideals into a solution of the Gauss Problem4 (Cf. Weber310). He reproduced and amplified (pp. 269-541) Klein's theory of the modular function.134 He denoted (p. 360) by W the substitution w'=-w/n and by rTl(n) that sub-group of the modular group o'=(aw+/)/(yw+8) for which y7O (mod n). The fundamental polygon134 for the group r~ (n) is called the transformation polygon Tn. Fricke found (p. 363) that in T,, the number of fixed points for elliptic substitutions of period 2 among the substitutions of r~(n).W is Cl(-4n) if n=O, 1, 2 (mod 4) and is Cl(-4n) + C (-n) if n-3 (mod 4). Finally it should be noted that the class-number may be deduced373 from the number of classes of ideals in an algebraic field since there is a (1, 1) correspondence between the classes of binary quadratic forms of discriminant D and the narrow classes of ideals in a quadratic field of discriminant D (Dedekind29 of Ch. III). For the class-number of forms with complex integral coefficients, see Ch. VIII. 372 Amer. Jour. Math., Jan., 1923. Abstract in Records of Proceedings of London Math. Soc., Nov. 17, 1921. 373 Dedekind in Dirichlet's Zahlentheorie, ed. 4, 1894, 639. 374 Annals of Math., 23, March, 1922. 375 Die Elliptischen Functionen und ihre Anwendungen, II, 1922. AUTHOR INDEX. The numbers refer to pages. Those in parenthesis relate to cross-references. CI. VI. NUMBER OF CLASSES OF BINARY QUADRATIC FORMS WITH INTEGRAL COEFFICIENTS. Aladow, J. S., 158-9 Appell, P., 114, 160 Arndt, F., 107 (102, 117, 151) Bachmann, P., 151, 183 (116, 120, 123) Bell, E. T., 114, 194-7 (115, 160, 197) Berger, A., 132-3, 136-7 (111) Biehler, C., 114, 188 Byerly, W. E., 98 Cauchy, A. L., 102-3 (124, 169, 182) Cayley, A., 131 Chapelon, J., 185,186-7 (117, 121-2) Chasles, M., 131-2 Chatelain, E., 181 (95) Clausen, C., 145 Cresse, G. H., 197 (185) Dedekind, R., 92, 122, 125-6, 159 (95-7, 119, 149, 150, 165, 172, 179, 197) De la Vallee Poussin, Ch., 188 De Seguier, J., 140, 151-3, 158 (95, 139) Dickson, L. E., 183-4 (95) Dirichlet, G. L., 92, 97-102, 104, 117 (95, 102-3, 105, 107, 109, 111, 115, 118-9, 122, 124, 129, 136-40, 146 -8, 150-1, 153-4, 156-9, 162, 164-7, 169-70, 172-5, 178-9, 181-2, 188, 192, 195) Eisenstein, G., 103 Erler, H. W., 103-4 Euler, L., 184 (146, 164) Fourier, J. Bapt. J., 101 Friedmann, A., 182 (102) Fricke, R., 130, 137, 150, 197 (106, 142, 177) Gauss, C. F., 92-7, 103, 151 (94, 98-100, 104-5, 109, 111, 115, 120, 123-5, 127, 134-5, 143, 151-2, 155, 163, 171. 178-9, 181, 197) Gegenbauer, L., 143-4, 146-7, 155-6 (134) Gierster, J., 92, 127-8, 130-2 (117, 128, 138, 140-2, 150, 177, 185-7) Glaisher, J. W. L., 164-5 Goring, W., 188 Gbtting, R., 122, 154 Hacks, J., 150 Haussner, R., 99 Hermite, C., 92-3, 106-7, 111-5, 133, 135-6, 145-8 (109-10, 113, 116-7, 120-2, 124-6, 134, 143, 148, 157, 160, 169, 174, 182, 187, 191, 194, 196-7) Holden, H., 171-4, 179 (118) Humbert, G., 92-4, 174-8, 180-2, 185-6, 191-3 (98, 100, 109, 114, 117, 120-2, 161, 185, 187-9, 191) Hurwitz, A., 92, 128-9, 137-8, 140-2, 144-5, 154-5, 167-8 (117, 128, 136, 142-3, 150, 162, 175, 177, 179) Jacobi, C. G. J., 111, 114, 145, 160, 166 (98, 103, 113) Jordan, C., 160 Joubert, P., 109-11 (95, 107, 120) Karpinski, L., 165 (172) Klein, F., 92, 126-8, 130, 137, 141, 150 (106, 127-8, 131-2, 141-2, 148, 150, 177, 197) Koenigsberger, L., 106 Kronecker, L., 92-3, 105-6, 108-9, 113-4, 119-20, 124-5, 127, 129-30, 138-40, 148 (93, 95, 100, 106, 109, 111-5, 117, 120, 127-30,134-7, 139 -40, 143-4, 149, 151-3, 156, 160, 163, 165, 168-70, 175, 178-83, 185-6, 191-2, 194-5, 197) Landau, E., 163, 165, 184-5, 188, 192 (95, 143, 146, 183) Lebesgue, V. A., 103 (97, 118, 151, 155, 169, 171, 179) Legendre, A. M., 94 (114) Lehmer, D. N., 159-60 (159) Lerch, M., 92, 124, 153-8, 162-4, 166, 168-71, 181-4 (95, 100, 103, 151, 179, 181, 183) Levy, F., 186 Liouville, J., 92, 115-7, 120-2, 133 (117, 120, 134, 160-1, 174-5, 183, 187, 190, 197) Lipschitz, R., 104-5, 120 (95, 100, 109, 114, 123, 134, 146, 150-1, 181) Mathews, G. B., 150 (98, 104-5) Meissner, E., 174 (116, 183) Mertens, F., 123, 155 (95, 134, 143-4, 151, 155) Moebius, A. F., (123, 147, 152, 169) Mordell, L. J., 113, 188, 190 -5, 197 (115, 160) Nagel, T., 186 (95) Nazimow, P., 148 Neumann, C., 140 Osborn, G., 154 Pepin, T., 122-4,127 (95,103, 115, 117) Petr, K., 92, 160-3, 178-9, 181, 188-90 (114, 117, 163, 175, 177, 188, 191) Pexider, J. V., 181-2 (98) Pfeiffer, E., 143 (146, 184) Picard E., 141 Plancherel, M., 179-80 Poincare, H., 141, 166-7 (117, 178) Rabinovich, G., 186 (95) Riemann, B., 140, 141 Schemmel, V., 117-9 (151, 172, 181) Selling, E, 160 Smith, H. J. S., 111, 114, 116, 120, 126 (104-6,108-9, 111 -2, 125-7, 150) Stern, M. A., 103, 182 (182) Stieltjes, T. J., 94, 133-6 (117, 122) Stouff, X., 148 Tamarkine, J., 182 (102) Teege, H., 178 Uspensky, J. V., 185 (197) Weber, H., 139-40, 148-52, 179 (95, 106, 150, 155-6, 158, 178, 181, 197) Wright, H. N., 187-8