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arXiv · 1610.05840

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

Bruce C. Berndt·Atul Dixit·Sun Kim·Alexandru Zaharescu
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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.

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Bruce C. Berndt, Atul Dixit, Sun Kim, Alexandru Zaharescu. 2016-10-19. On a theorem of A. I. Popov on sums of squares. https://arxiv.org/abs/1610.05840

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