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

Access Restriction
Authorized

 Author Dixit, Atul ♦ Zaharescu, Alexandru ♦ Kim, Sun Advisor Berndt, Bruce C. Source IIT Gandhinagar Content type Text Publisher American Mathematical Society Language English
 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. ISSN 10886826 Learning Resource Type Article Publisher Date 2017-04-01 e-ISSN 00029939 Journal Proceedings of the American Mathematical Society Volume Number 145 Issue Number 9 Starting Page 3795 Ending Page 3808