Contributions to the Theory of Zeta-Functions

The Modular Relation Supremacy

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