CONTRIBUTIONS TO THE THEORY OF ZETA-FUNCTIONS: The Modular Relation Supremacy

by Shigeru Kanemitsu, Haruo Tsukada

Overview

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.

Product Details

ISBN-13:	9789814449632
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/15/2014
Series:	SERIES ON NUMBER THEORY AND ITS APPLICATIONS , #10
Sold by:	Barnes & Noble
Format:	eBook
Pages:	316
File size:	27 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

1 Prelude 1

1.1 Introduction 1

1.2 Eternal return or every 50 years 12

1.2.1 Poincaré recurrence theorem 12

1.2.2 Lerch-Chowla-Selberg formula 12

1.2.3 Knopp-Hasse-Sondow formula 15

1.3 The theta-transformation formula 20

1.4 Summation formulas 23

1.4.1 Poisson summation formula 23

1.4.2 Generalization of the Plana summation formula 25

2 Grocery of Special Functions 29

2.1 Formulas for the ganuna function and their use 29

2.2 Zeta-functions 33

2.3 Bessel functions 35

2.4 Ψ-functions 38

2.5 Generalized hypergeometric functions 39

2.6 Fox H-functions 49

2.7 Formulas for Fox and Meijer functions 51

2.8 Special cases of G-functions 56

3 Unprocessed Modular Relations 61

3.1 The H^M,0_0,M ↔ H^N,0_0,N formula 61

3.1.1 The H^1,0_0,1 ↔ H^1,0_0,1 formula 66

3.1.2 The G^2,0_0,2 ↔ G^2,0_0,2 formula 69

3.2 Dedekind zeta-function I 70

3.3 Transformation formulas for Lambert series 72

3.3.1 Lambert series 73

3.3.2 Lambert series and short character sums 78

3.3.3 Ramanujan's formula leading to the eta-transformation formula 80

3.3.4 A brief account of modular forms 84

3.3.5 The Ramanujan-Guinand formula 86

3.3.6 The reciprocity law for Dedekind sums 93

3.4 Koshlyakov's method [Koshl] 95

3.4.1 Dedekind zeta-function II 96

3.5 Koshlyakov's functions 97

3.5.1 Koshlyakov's X-functions 97

3.5.2 Koshlyakov's V-function 101

4 Fourier-Bessel Expansion ^1,1_1,1 ↔ H^2,0_0,2107

4.1 Introduction 107

4.2 Stark's method 108

4.2.1 Murty-Sinha theorem 108

4.2.2 Stark's method 111

4.3 The main formula for modular relations 115

4.3.1 Specification of Theorem 4.4 119

4.4 Dedekind zeta-function III 122

4.5 Elucidation of Koshlyakov's result in the real quadratic case 127

4.6 Koshlyakov's K-series 128

4.7 The Fourier-Bessel expansion ^1,1_1,1 ↔ G^2,0_0,2 133

4.8 Bochner-Chandrasekharan and Narasimhan formula 142

5 The Ewald Expansion or the Incomplete Gannua Series 145

5.1 Ewald expansion for zeta-functions with a single gamma factor 145

5.1.1 Confluent hypergeometric series imply incomplete gamma series, Ewald expansions 148

5.1.2 Bochner-Chandrasekharan formula as H^2,0_1,2 ↔ H^1,1_1,2 150

5.2 Atkinson-Berndt Abel mean 154

5.2.1 Landau's exposition 156

5.2.2 Screened Coulomb potential 158

6 The Riesz Sums 169

6.1 Various modular relations 169

6.1.1 Riesz sums 169

6.1.2 Improper modular relations as Riesz sums 172

6.1.3 The H^1,1_2,1 ↔ H^2,0_1,3 formula 173

6.1.4 The H^1,1_2,2 ↔ H^2,0_1,3 formula 174

6.1.5 Katsurada's formula, combined 175

6.1.6 Linearized product of two zeta-functions 178

6.2 Modular relations in integral form 185

6.2.1 Integration in the parameter 185

6.2.2 Generalization of Ramanujan's integral formula 188

6.3 Integrated modular relations 191

6.3.1 The Hardy-Littlewood sum 192

6.3.2 The H^1,1_2,1 ↔ H^2,0_0,3 formula 194

6.3.3 Arithmetical Fourier series 196

6.3.4 Riemami's legacy 206

7 The General Modular Relation 211

7.1 Definitions 211

7.2 Assumptions 213

7.3 Theorem 216

7.4 The Main Formula (basic version) 220

8 The Hecke Type Zeta-functions 225

8.1 Statement of the formula 225

8.1.1 The bilateral form 227

8.1.2 The Bochner modular relation: G^1,0_0,1 ↔ G^1,0_0,1 229

8.2 The Riesz sums or the first J-Bessel expansion: G^1,0_1,1 ↔ G^1,0_0,2 229

8.3 The partial sum formula: G^2,0_2,2 ↔ G^1,1_1,3 230

8.4 The Fourier-Bessel expansion: G^1,1_1,1 ↔ G^2,0_0,2 231

8.5 The Ewald expansion: G^2,0_1,2 ↔ G^1,1_1,2 231

8.6 The Bochner-Chandrasekharan formula: H^2,0_1,2 ↔ H^1,1_1,2232

8.7 The G^2,1_1,2 ↔ H^2,1_1,2 formula 232

8.8 The second J-Bessel expansion: G^1,1_2,2 ↔ G^2,0_1,3 232

8.9 The H^2,1_2,2 ↔ H^2,1_1,3 formula 234

8.10 The second K-Bessel expansion: G^3,0_1,3 ↔ G^1,2_2,2235

8.11 The G^3,1_1,3 ↔ G^2,2_2,2 formula 236

8.12 The G^2,1_2,3 ↔ G^2,1_2,3 formula 236

8.13 The G^3,0_2,3 ↔ G^1,2_2,3 formula 237

8.14 The G^3,1_2,3 ↔ G^3,2_2,3 formula 237

8.15 The G^3,2_2,3 ↔ G^3,2_2,3 formula 238

8.16 The G^p+1,0_1,p+1 ↔ G^1,p_p,2 formula 239

9 The Product of Zeta-functions 241

9.1 The product; of zeta-functions 241

9.1.1 Statement of the Main Formula 241

9.1.2 Wilton's Riesz sum: G^2,2_4,4 ↔ G^4,0_2,6 243

9.2 Powers of zeta-functions 253

9.2.1 Statement of the Main Formula 253

9.2.2 The G^N,0_N,N ↔ G^N,0_0,2N formula 257

9.2.3 The G^q+1,0_q+1,q+1 ↔ G^N,q-N+1_q-N+1,q+N+1 formula 263

10 Miscellany 267

10.1 Future projects 267

10.1.1 Rankin-Selberg convolution 267

10.1.2 Maass forms 269

10.1.3 G-functions of two variables 270

10.1.4 Plausible general form 273

10.2 Quellenangaben 275

10.2.1 Berudt-Knopp book and Berndt's series of papers 275

10.2.2 Corrections to "Number Theory and its Applications" 276

Bibliography 279

Index 301