Equivalents of the Riemann Hypothesis: Volume 2, Analytic Equivalents

Kevin Broughan

• Publisher: Cambridge University Press
• Year: 2017
• Print ISBN: 9781107197121
• eBook ISBN: 9781108195430
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

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.

Table of contents

• Cover
• Half-title
• Series page
• Title page
• Copyright information
• Dedication1
• Epigraph
• Contents for Volume Two
• Contents for Volume One
• List of illustrations
• List of tables
• Preface
• Acknowledgements
• 1 Introduction
• 2 Series Equivalents
• 3 Banach and Hilbert Space Methods
• 4 The Riemann Xi Function
• 5 The De Bruijn–Newman Constant
• 6 Orthogonal Polynomials
• 7 Cyclotomic Polynomials
• 8 Integral Equations
• 9 Weil's Explicit Formula, Inequality and Conjectures
• 10 Discrete Measures
• 11 Hermitian Forms
• 12 Dirichlet L-Functions
• 13 Smooth Numbers
• 14 Epilogue
• Appendix A Convergence of Series
• Appendix B Complex Function Theory
• Appendix C The Riemann–Stieltjes Integral
• Appendix D The Lebesgue Integral on R
• Appendix E The Fourier Transform
• Appendix F The Laplace Transform
• Appendix G The Mellin Transform
• Appendix H The Gamma Function
• Appendix I The Riemann Zeta Function
• Appendix J Banach and Hilbert Spaces
• Appendix K Miscellaneous Background Results
• Appendix L GRHpack Mini-Manual
• References
• Index