Arithmetical identities and zeta-functions

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 f...

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Bibliographic Details
Published in:Mathematische Nachrichten 2011-02, Vol.284 (2-3), p.287-297
Main Authors: Kanemitsu, Shigeru, Ma, Jing, Tanigawa, Yoshio
Format: Article
Language:English
Subjects:
11M26
11M41
arithmetic Fourier series
Bernoulli polynomials
Modular relation
MSC 11F66
MSC 11F66, 11M26, 11M41
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Summary: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 Möbius 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.200710212