Maihemalik Dedekind Sums and Class Numbers

Formulas for the class number of an imaginary quadratic number field are proved. Some of these formulas were previously established by BEI~NI)T and by GOLDSTEIN and RAzE with the use of analytic methods. The proofs given here use Dirichlet's classical class number formula, but otherwise the proofs are completely elementary. A key ingredient in the proofs is the reciprocity theorem for Dedekind--Rademacher sums. Throughout the sequel, M denotes a matrix with integral entries a,b,e, and d such that a d b c = 1; k denotes a natural number; and g denotes a character (mod k). Write Let F(/c) be the group of matrices M such tha t M----=LI(modk). Let G(/c) denote the group of matrices generated by S~ and T. Write A (k)~-G (k)(~ F(k) and set A (It)T : { M T : M e A (k)}. Given an imaginary quadratic number field of discriminant -k , let h(--/c) denote its class number and let w~ denote the number of roots of uni ty in this field. As usual, let ((x)) ---/ x [x]0,-1/2, ifotherwise.X is not an integer, l~onatshefte ffir 1Kathematik, Bd. 84/4 19 266 B.C. BERNDT and R. J. EvANs In 1976, GOLDS~EIN and RAZAR [5, p. 358] derived the following formula for h (--~)~: m(modak) n(modk) when M~A (k)T, c >0, and 1: is odd, real, and primitive. Their proof depends upon two different formulations of the transformation formulae for oo -'X(m) x(n) e2:~tmnz/k , (e) Ib ~ n , n ~ l where I m ( z ) > 0 . One of these formulations is a special case of a general theorem proved in 1973 by BERNDT [2, Theorems 2, 3]. See also [3, Theorem 1]. Moreover, using the transformation formulae for (2), but in a manner less complicated than that in [5], BEI~NDT [2, Theorem 4] essentially established the following class number formula: m(modck) n(modk) (3) § ]i ' m(modak) n(modk) where g is odd, real, and primitive, and where a and c are positive integers such that either k]a or k]c. Thus, (3) was established under less restrictions than (1). The formulation of (3) in [2] is in terms of Dedekind character sums and the generMized Bernoulli number B1(;%). However, by the use of Dirichlet's class number formula [6, p. 405] k-1 W k K-" h(--]c) : 9 ~ _ _ , g(n)n, (4) n = l it follows at once tha t Bl(z) : -2h(-k) /w~, and so the formulation (3) is easily seen to be equivalent to tha t in [2, Theorem 4]. In this paper, we prove by elementary methods that the second double sum on the right side of (3) vanishes whenever MeA (It)T and Z is odd, real, and primitive (see Theorem 7 and Lemma 2). Dedekind Sums and Class Numbers 267 We also give completely e lementary proofs of (1) and (3) (see Theorems 4 and 8 and L e m m a 2). I f h and k are integers with k > 0 and if x and y are real numbers , recall tha t the Dedek indI{ademacher sum s(h,k;x,y) is defined b y