An analogue of a formula of Popov

Abstract

Let $r_k(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a new summation formula involving $r_k(n)$ and the Bessel functions of the first kind, which constitutes an analogue of a result due to the Russian mathematician A. I. Popov.

V2: We have added a final section containing a proof of a generalization of the Ramanujan-Guinand formula; Reference [13] of the previous version corrected