Equivalents of the Riemann Hypothesis: Volume 2, Analytic Equivalents (Encyclopedia of Mathematics and its Applications, Series Number 165)

Cover 1 Half-title 3 Series page 4 Title page 5 Copyright information 6 Dedication1 7 Epigraph 8 Contents for Volume Two 9 Contents for Volume One 13 List of illustrations 16 List of tables 18 Preface 19 Acknowledgements 22 1 Introduction 23 1.1 Why This Study? 23 1.2 Summary of Volume Two 24 1.3 How to Read This Book 29 2 Series Equivalents 30 2.1 Introduction 30 2.2 The Riesz Function 32 2.3 Additional Properties of the Riesz Function 36 2.4 The Series of Hardy and Littlewood 37 2.5 A General Theorem for a Class of Entire Functions 38 2.6 Further Work 44 3 Banach and Hilbert Space Methods 45 3.1 Introduction 45 3.2 Preliminary Definitions and Results 47 3.3 Beurling's Theorem 51 3.4 Recent Developments 57 4 The Riemann Xi Function 59 4.1 Introduction 59 4.2 Preliminary Results 62 4.3 Monotonicity of |ξ(s)| 71 4.4 Positive Even Derivatives 73 4.5 Li's Equivalence 76 4.6 More Recent Results 81 5 The De Bruijn–Newman Constant 84 5.1 Introduction 84 5.2 Preliminary Definitions and Results 88 5.3 A Region for Ξλ(z) With Only Real Zeros 91 5.4 The Existence of Λ 99 5.5 Improved Lower Bounds for Λ 99 5.5.1 Lehmer's Phenomenon 100 5.5.2 The Differential Equation Satisfied by H(t,z) 103 5.5.3 Finding a Lower Bound for ΛC Using Lehmer Pairs 109 5.6 Further Work 114 6 Orthogonal Polynomials 115 6.1 Introduction 115 6.2 Definitions 116 6.3 Orthogonal Polynomial Properties 118 6.4 Moments 121 6.5 Quasi-Analytic Functions 126 6.6 Carleman's Inequality 128 6.7 Riemann Zeta Function Application 135 6.8 Recent Work 138 7 Cyclotomic Polynomials 139 7.1 Introduction 139 7.2 Definitions 140 7.3 Preliminary Results 141 7.4 Riemann Hypothesis Equivalences 146 7.5 Further Work 148 8 Integral Equations 149 8.1 Introduction 149 8.2 Preliminary Results 151 8.3 The Method of Sekatskii, Beltraminelli and Merlini 155 8.4 Salem's Equation 161 8.5 Levinson's Equivalence 164 9 Weil's Explicit Formula, Inequality and Conjectures 172 9.1 Introduction 172 9.2 Definitions 174 9.3 Preliminary Results 174 9.4 Weil's Explicit Formula 176 9.5 Weil's Inequality 181 9.6 Bombieri's Variational Approach to RH 188 9.7 Introduction to the Weil Conjectures 195 9.8 History of the Weil Conjectures 196 9.9 Finite Fields 198 9.10 The Weil Conjectures for Varieties 200 9.11 Elliptic Curves 200 9.12 Weil Conjectures for Elliptic Curves – Preliminary Results 204 9.13 Proof of the Weil Conjectures for Elliptic Curves 208 9.14 General Curves Over Fq and Applications 210 9.15 Return to the Explicit Formula 212 9.16 Weil's Commentary on His 1952 and 1972 Papers 214 10 Discrete Measures 215 10.1 Introduction 215 10.2 Definitions 216 10.3 Preliminary Results 217 10.4 A Mellin-Style Transform 219 10.5 Verjovsky's Theorems 222 10.6 Historical Development of Non-Euclidean Geometry 228 10.7 The Hyperbolic Upper Half Plane H 230 10.8 The Groups PSL(2,R) and PSL(2,Z) 231 10.9 Eisenstein Series 233 10.10 Zagier's Horocycle Equivalence 238 10.11 Additional Results 241 11 Hermitian Forms 243 11.1 Introduction 243 11.2 Definitions 245 11.3 Distributions 248 11.4 Positive Definite 250 11.5 The Restriction to C(a) for All a>0 253 11.6 Properties of K(a) and K(a) 258 11.7 Matrix Elements 264 11.8 An Explicit Example With a = log2 269 11.9 Lemmas for Yoshida's Main Theorem 280 11.10 Hermitian Forms Lemma 282 11.11 Yoshida's Main Theorem 291 11.12 The Restriction to K(a) for All a>0 292 12 Dirichlet L-Functions 296 12.1 Introduction 296 12.2 Definitions 299 12.3 Properties of L(s,χ) 305 12.4 The Non-Vanishing of L(1,χ) 306 12.5 Zero-Free Regions and Siegel Zeros 310 12.6 Preliminary Results for Titchmarsh's Criterion 317 12.7 Titchmarsh's GRH Equivalence 318 12.8 Preliminary Results for Gallagher's Theorem 320 12.9 Gallagher's Theorems 324 12.10 Applications of Gallagher's Theorems 329 12.11 The Bombieri–Vinogradov Theorem 333 12.12 Applications of GRH Bombieri–Vinogradov's Theorem 345 12.13 Generalizations and Developments for Bombieri–Vinogradov 348 12.14 Conjectures 349 13 Smooth Numbers 354 13.1 Introduction 354 13.2 The Dickman Function 357 13.3 Preliminary Lemmas for Hildebrand's Equivalence 368 13.4 Riemann Hypothesis Equivalence 371 13.5 Further Work 379 14 Epilogue 381 Appendix A Convergence of Series 383 Appendix B Complex Function Theory 385 Appendix C The Riemann–Stieltjes Integral 399 Appendix D The Lebesgue Integral on R 403 Appendix E The Fourier Transform 410 Appendix F The Laplace Transform 427 Appendix G The Mellin Transform 431 Appendix H The Gamma Function 440 Appendix I The Riemann Zeta Function 447 Appendix J Banach and Hilbert Spaces 464 Appendix K Miscellaneous Background Results 473 Appendix L GRHpack Mini-Manual 481 L.1 Introduction 481 L.1.1 Installation 481 L.1.2 About This Mini-Manual 482 L.2 GRHpack Functions 483 References 495 Index 507 "The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proof."--Back cover The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proofs. This two-volume work presents the main known equivalents to the Riemann hypothesis, perhaps the most important problem in mathematics. Volume 2 covers equivalents with a strong analytic orientation and is supported by an extensive set of appendices. Volume 1. Arithmetic Equivalents -- Volume 2. Analytic Equivalents. Kevin Broughan, University Of Waikato, New Zealand. Includes Bibliographical References And Indexes.