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

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

doi:10.1090/proc/13547

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

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

Department of Mathematics

University of Illinois at Urbana-Champaign
Indian Institute of Technology Gandhinagar
Romanian Academy

Research output: Contribution to journal › Journal article › peer-review

6 Scopus citations

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.

Original language	English
Pages (from-to)	3795-3808
Number of pages	14
Journal	Proceedings of the American Mathematical Society
Volume	145
Issue number	9
DOIs	https://doi.org/10.1090/proc/13547
State	Published - 2017

Keywords

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

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10.1090/proc/13547

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title = "On a theorem of A. I. Popov on sums of squares",

abstract = "Let rk (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 rk (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{\textquoteright}s identity and an identity involving r2(n) from Ramanujan{\textquoteright}s lost notebook.",

keywords = "Bessel functions, Dirichlet characters, Dirichlet series, Ramanujan{\textquoteright}s lost notebook, Sums of squares, Vorono{\"i} summation formula",

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TY - JOUR

T1 - On a theorem of A. I. Popov on sums of squares

AU - Berndt, Bruce C.

AU - Dixit, Atul

AU - Kim, Sun

AU - Zaharescu, Alexandru

PY - 2017

Y1 - 2017

N2 - Let rk (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 rk (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.

AB - Let rk (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 rk (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.

KW - Bessel functions

KW - Dirichlet characters

KW - Dirichlet series

KW - Ramanujan’s lost notebook

KW - Sums of squares

KW - Voronoï summation formula

UR - https://www.scopus.com/pages/publications/85021435530

U2 - 10.1090/proc/13547

DO - 10.1090/proc/13547

M3 - Journal article

AN - SCOPUS:85021435530

SN - 0002-9939

VL - 145

SP - 3795

EP - 3808

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 9

ER -

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

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Keywords

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