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. Book jacket cover 1 Title 4 Preface 8 Contents 10 Chapter 1 Prelude 14 Chapter 2 Grocery of Special Functions 42 Chapter 3 Unprocessed Modular Relations 74 Chapter 4 Fourier-Bessel Expansion H1;11;1$ H2;00;2 120 Chapter 5 The Ewald Expansion or the Incomplete Gamma Series 158 Chapter 6 The Riesz Sums 182 Chapter 7 The General Modular Relation 224 Chapter 8 The Hecke Type Zeta-functions 238 Chapter 9 The Product of Zeta-functions 254 Chapter 10 Miscellany 280 Bibliography 292 Index 314