Weighted divisor sums and Bessel function series, IV

Bruce C. Berndt; Sun Kim; Alexandru Zaharescu

doi:10.1007/s11139-011-9352-8

Weighted divisor sums and Bessel function series, IV

Bruce C. Berndt, Sun Kim, Alexandru Zaharescu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

One fragment (p. 335) published with Ramanujan's Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.

Original language	English (US)
Pages (from-to)	79-102
Number of pages	24
Journal	Ramanujan Journal
Volume	29
Issue number	1-3
DOIs	https://doi.org/10.1007/s11139-011-9352-8
State	Published - Dec 2012

Keywords

Bessel functions
Circle problem
Dirichlet L-series
Ramanujan's Lost Notebook
Riesz sums
Weighted divisor sums

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.1007/s11139-011-9352-8

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title = "Weighted divisor sums and Bessel function series, IV",

abstract = "One fragment (p. 335) published with Ramanujan's Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.",

keywords = "Bessel functions, Circle problem, Dirichlet L-series, Ramanujan's Lost Notebook, Riesz sums, Weighted divisor sums",

author = "Berndt, {Bruce C.} and Sun Kim and Alexandru Zaharescu",

note = "Funding Information: The first author{\textquoteright}s research was partially supported by NSA Grant H98230-11-1-0200. The third author{\textquoteright}s research was partially supported by NSF Grant DMS-0901621.",

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AU - Berndt, Bruce C.

AU - Kim, Sun

AU - Zaharescu, Alexandru

N1 - Funding Information: The first author’s research was partially supported by NSA Grant H98230-11-1-0200. The third author’s research was partially supported by NSF Grant DMS-0901621.

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N2 - One fragment (p. 335) published with Ramanujan's Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.

AB - One fragment (p. 335) published with Ramanujan's Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.

KW - Bessel functions

KW - Circle problem

KW - Dirichlet L-series

KW - Ramanujan's Lost Notebook

KW - Riesz sums

KW - Weighted divisor sums

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Weighted divisor sums and Bessel function series, IV

Abstract

Keywords

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