paper

On a theorem of A. I. Popov on sums of squares

arXiv:1610.05840

Abstract

Let $r_{k} (n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. In 1934, the Russian mathematician A.~I.~Popov stated, but did not rigorously prove, a beautiful series transformation involving $r_{k} (n)$ and certain Bessel functions. We provide a proof of this identity for the first time, as well as for another identity, which can be regarded as both an analogue of Popov's identity and an identity involving $r_{2} (n)$ from Ramanujan's lost notebook.

14 pages, to appear in Proc. Amer. Math. Soc