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Arithmetical identities and zeta‐functions
Author(s) -
Kanemitsu Shigeru,
Ma Jing,
Tanigawa Yoshio
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710212
Subject(s) - arithmetic function , mathematics , riemann hypothesis , modular form , fourier series , pure mathematics , riemann zeta function , bernoulli's principle , fourier transform , differentiable function , series (stratigraphy) , polynomial , algebra over a field , mathematical analysis , biology , aerospace engineering , paleontology , engineering
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