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


Authors: Bruce C. Berndt, Atul Dixit, Sun Kim and Alexandru Zaharescu
Journal: Proc. Amer. Math. Soc. 145 (2017), 3795-3808
MSC (2010): Primary 11E25; Secondary 33C10
DOI: https://doi.org/10.1090/proc/13547
Published electronically: April 7, 2017
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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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Additional Information

Bruce C. Berndt
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: berndt@illinois.edu

Atul Dixit
Affiliation: Department of Mathematics, Indian Institute of Technology, Gandhinagar, Palaj, Gandhinagar 382355, Gujarat, India
Email: adixit@iitgn.ac.in

Sun Kim
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: sunkim2@illinois.edu

Alexandru Zaharescu
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801 – and – Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO–014700 Bucharest, Romania
Email: zaharesc@illinois.edu

DOI: https://doi.org/10.1090/proc/13547
Keywords: Sums of squares, Bessel functions, Vorono\"{\i} summation formula, Dirichlet series, Dirichlet characters, Ramanujan's lost notebook
Received by editor(s): October 18, 2016
Published electronically: April 7, 2017
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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