On the summation of the singular series (Q1091424)

This paper deals with explicit evaluation, in certain cases, of \(r(n, \mathrm{gen}\,I_k)\) which denotes the average number of representations of \(n\) by the genus of \(I_k\), the identity quadratic form in \(k\) variables. The methods rely heavily on Siegel's formula for this quantity in terms of \(p\)-adic densities. The author uses Gauss sums to evaluate the 2-adic density in the cases she considers. She obtains the classical results of Ramanujan for sums of 24 squares and the results of Bateman when \(k=3\) or 4.