On the summation of the singular series

Abstract

The singular series has great importance in the study of the number of representations of a rational integer as a sum oí integral squares (cf. (3)). As is known (cf. (3), [81) the sum of the singular series is jusi the average number r(n, gen Jk) of representations of a positive integer n by the genus of the identity quadralic form in k variables. Bateman [2) calculated the sum of the singular series in the cases k = 3.4, following Hardy and Hecke methocl~. His results can though more easily be obtained by using Siegel's formula for the evaluation of r(n, gen lk). In this paper we derive in sorne special cases a formula for r(n, gen lk)• from Siegel's formula. which covers those considered by Bateman. We use Gauss-Weber sums to evaluate the 2-adic densities, which, in Siegel's rnethod, causes the main difficulties. Our considerations in the case k • 24 also yield the celebrated Ramanujan's formula about the number of representations of an integer as sum of 24 squares. I wish to thank Professor P. Bayer for her encouragement in doing this paper.