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

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

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)3795-3808
Number of pages14
JournalProceedings of the American Mathematical Society
Volume145
Issue number9
DOIs
StatePublished - 2017

Keywords

  • Bessel functions
  • Dirichlet characters
  • Dirichlet series
  • Ramanujan’s lost notebook
  • Sums of squares
  • Voronoï summation formula

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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