Identities involving the coefficients of a class of dirichlet series. vii

Bruce C. Berndt

doi:10.1090/S0002-9947-1975-0352018-5

Identities involving the coefficients of a class of dirichlet series. vii

Bruce C. Berndt

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

Abstract

Let a(n) be an arithmetical function, and consider the Riesz sum Ar(x) = S_n£_xa(n)(x-n)^r. For a(n) belonging to a certain class of arithmetical functions, Ar(x) can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of r. Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.

Original language	English (US)
Pages (from-to)	247-261
Number of pages	15
Journal	Transactions of the American Mathematical Society
Volume	201
DOIs	https://doi.org/10.1090/S0002-9947-1975-0352018-5
State	Published - 1975
Externally published	Yes

Keywords

Arithmetical function
Average order of arithmetical functions
Bessel function identity
Dirichlet series
Functional equation involving gamma factors
Hardy-Landau circle method
Riesz sum

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

Online availability

10.1090/S0002-9947-1975-0352018-5

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abstract = "Let a(n) be an arithmetical function, and consider the Riesz sum Ar(x) = Sn£xa(n)(x-n)r. For a(n) belonging to a certain class of arithmetical functions, Ar(x) can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of r. Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.",

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PY - 1975

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AB - Let a(n) be an arithmetical function, and consider the Riesz sum Ar(x) = Sn£xa(n)(x-n)r. For a(n) belonging to a certain class of arithmetical functions, Ar(x) can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of r. Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.

KW - Arithmetical function

KW - Average order of arithmetical functions

KW - Bessel function identity

KW - Dirichlet series

KW - Functional equation involving gamma factors

KW - Hardy-Landau circle method

KW - Riesz sum

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JO - Transactions of the American Mathematical Society

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ER -

Identities involving the coefficients of a class of dirichlet series. vii

Abstract

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