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Arithmetical identities and zeta-functions

✍ Scribed by Shigeru Kanemitsu; Jing Ma; Yoshio Tanigawa

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
128 KB
Volume
284
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

In this paper we establish a class of arithmetical Fourier series as a manifestation of the intermediate modular relation, which is equivalent to the functional equation of the relevant zeta‐functions. One of the examples is the one given by Riemann as an example of a continuous non‐differentiable function. The novel interest lies in the relationship between important arithmetical functions and the associated Fourier series. E.g., the saw‐tooth Fourier series is equivalent to the corresponding arithmetical Fourier series with the MΓΆbius function. Further, if we squeeze out the modular relation, we are led to an interesting relation between the singular value of the discontinuous integral and the modification summand of the first periodic Bernoulli polynomial. Β© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

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