Contributions to the Theory of Zeta-Functions

The Modular Relation Supremacy

Nonfiction, Science & Nature, Mathematics, Functional Analysis, Number Theory
Cover of the book Contributions to the Theory of Zeta-Functions by Shigeru Kanemitsu, Haruo Tsukada, World Scientific Publishing Company
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Author: Shigeru Kanemitsu, Haruo Tsukada ISBN: 9789814449632
Publisher: World Scientific Publishing Company Publication: December 15, 2014
Imprint: WSPC Language: English
Author: Shigeru Kanemitsu, Haruo Tsukada
ISBN: 9789814449632
Publisher: World Scientific Publishing Company
Publication: December 15, 2014
Imprint: WSPC
Language: English

This volume provides a systematic survey of almost all the equivalent assertions to the functional equations — zeta symmetry — which zeta-functions satisfy, thus streamlining previously published results on zeta-functions. The equivalent relations are given in the form of modular relations in Fox H-function series, which at present include all that have been considered as candidates for ingredients of a series. The results are presented in a clear and simple manner for readers to readily apply without much knowledge of zeta-functions.

This volume aims to keep a record of the 150-year-old heritage starting from Riemann on zeta-functions, which are ubiquitous in all mathematical sciences, wherever there is a notion of the norm. It provides almost all possible equivalent relations to the zeta-functions without requiring a reader's deep knowledge on their definitions. This can be an ideal reference book for those studying zeta-functions.

Contents:

  • Prelude

  • Grocery of Special Functions

  • Unprocessed Modular Relations

  • Fourier-Bessel Expansion H1;11;1↔ H2;00;2

  • The Ewald Expansion or the Incomplete Gamma Series

  • The Riesz Sums

  • The General Modular Relation

  • The Hecke Type Zeta-functions

  • The Product of Zeta-functions

  • Miscellany

  • Readership:

  • Key Features:

  • The book gives a thorough description of a.a. possible equivalent relations to the functional equation in terms of the H-function series for zeta-functions that have been studied

    Given a functional equation, the book immediately informs the reader of the direction to be taken to reach the ultimate aim concerning the zeta-function in both pure and applied disciplines

    The reader can forget about the difficult definition of zeta-functions themselves and can refer to the form of functional equation and check how multiple is the gamma factor

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This volume provides a systematic survey of almost all the equivalent assertions to the functional equations — zeta symmetry — which zeta-functions satisfy, thus streamlining previously published results on zeta-functions. The equivalent relations are given in the form of modular relations in Fox H-function series, which at present include all that have been considered as candidates for ingredients of a series. The results are presented in a clear and simple manner for readers to readily apply without much knowledge of zeta-functions.

This volume aims to keep a record of the 150-year-old heritage starting from Riemann on zeta-functions, which are ubiquitous in all mathematical sciences, wherever there is a notion of the norm. It provides almost all possible equivalent relations to the zeta-functions without requiring a reader's deep knowledge on their definitions. This can be an ideal reference book for those studying zeta-functions.

Contents:

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