TY - JOUR

T1 - Weighted divisor sums and Bessel function series, II

AU - Berndt, Bruce C.

AU - Kim, Sun

AU - Zaharescu, Alexandru

N1 - Funding Information:
* Corresponding author. E-mail addresses: berndt@illinois.edu (B.C. Berndt), kim.1674@math.ohio-state.edu (S. Kim), zaharesc@illinois.edu (A. Zaharescu). 1 The first author’s research was partially supported by NSA grant MDA904-00-1-0015. 2 The third author’s research was partially supported by NSF grant DMS-0901621.

PY - 2012/2/15

Y1 - 2012/2/15

N2 - On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively.

AB - On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively.

KW - Bessel functions

KW - Circle problem

KW - Divisor problem

KW - Fourier series

KW - Ramanujan's lost notebook

KW - Trigonometric sums

KW - Weighted divisor sums

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U2 - 10.1016/j.aim.2011.10.016

DO - 10.1016/j.aim.2011.10.016

M3 - Article

AN - SCOPUS:84855571287

VL - 229

SP - 2055

EP - 2097

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 3

ER -