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Transactions of the American Mathematical Society

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Identities involving the coefficients of a class of Dirichlet series. VII

Author: Bruce C. Berndt
Journal: Trans. Amer. Math. Soc. 201 (1975), 247-261
MSC: Primary 10H10; Secondary 30A16
MathSciNet review: 0352018
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Abstract: Let $ a(n)$ be an arithmetical function, and consider the Riesz sum $ {A_\rho }(x) = {\Sigma _{n \leq x}}a(n){(x - n)^\rho }$. For $ a(n)$ belonging to a certain class of arithmetical functions, $ {A_\rho }(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 $ \rho $. 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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Keywords: Dirichlet series, arithmetical function, functional equation involving gamma factors, Bessel function identity, Hardy-Landau circle method, average order of arithmetical functions, Riesz sum
Article copyright: © Copyright 1975 American Mathematical Society