This book presents a rich, insightful exploration of the interplay between randomness, the Riemann Hypothesis, and analytic number theory. Written for readers with a moderate background in mathematics (preferably having some familiarity with probability theory and complex analysis), it offers a deeper and more comprehensive account of the Riemann Hypothesis and its far-reaching consequences than is typically available in books aimed at general audiences. The book takes a relaxed and engaging approach, highlighting the beauty and intuition behind the key ideas. Technical proofs are kept to a minimum, with accessible arguments presented fully, while deeper results are described in context and supported by historical background and comprehensive references. This combination allows readers to appreciate the elegance of the subject and to navigate their own path toward more advanced study. Bringing together a wealth of material that is otherwise widely scattered across the literature, the book serves as an indispensable reference for researchers and enthusiasts alike. Throughout, it illuminates the crucial role that randomness plays in modern number theory-inspiring conjectures, guiding proofs, and opening new directions of inquiry. Six valuable appendices round out the volume, providing opportunities to review-or learn-essential background material across a range of mathematical topics.

Table of Contents

Chapter 1. Introduction

Chapter 2. The Prime Numbers

Chapter 3. Interlude on Arithmetic Functions

Chapter 4. The Heuristic of Harald Cramer

Chapter 5. The Zeta Function

Chapter 6. Statistics of the Zeros

Chapter 7. Statistics of the Critical Line

Chapter 8. Million Dollar Questions

Chapter 9. Frontiers of Chance and Chaos

Chapter 10. The Zeta Distributions

Chapter 11. Random Multiplicative Functions

Chapter 12. Further Topics

Appendix A. Analytic Functions

Appendix B. The Gamma and Related Functions

Appendix C. Dirichlet Series

Appendix D. The Statistical Limit Laws

Appendix E. Approximations for Sums and Products in Powers of n, p, logn, and logp

Appendix F. Some Mathematical Points

About the Authors

Andrey Feuerverger, University of Toronto, Canada

Greg Martin, University of British Columbia, Vancouver, Canada