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

2016/10/19 by Bruce C. Berndt, Berndt, Bruce C., Atul Dixit +5
Mathematics · #11E25 #33C10 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1610.05840

openalex publication_date 2016/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 r₂(n) from Ramanujan's lost notebook.

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