Bounded Gaps Between Primes: The Epic Breakthroughs of the Early Twenty-First Century

Description

Searching for small gaps between consecutive primes is one way to approach the twin primes conjecture, one of the most celebrated unsolved problems in number theory. This book documents the remarkable developments of recent decades, whereby an upper bound on the known gap length between infinite numbers of consecutive primes has been reduced to a tractable finite size. The text is both introductory and complete: the detailed way in which results are proved is fully set out and plenty of background material is included. The reader journeys from selected historical theorems to the latest best result, exploring the contributions of a vast array of mathematicians, including Bombieri, Goldston, Motohashi, Pintz, Yildirim, Zhang, Maynard, Tao and Polymath8. The book is supported by a linked and freely-available package of computer programs. The material is suitable for graduate students and of interest to any mathematician curious about recent breakthroughs in the field. Contents Preface Acknowledgements 1. Introduction 1.1 Why This Study? 1.2 Summary of This Chapter 1.3 History and Overview of These Developments 1.4 Polymath Projects and Members of Polymath8 1.5 Timeline of Developments 1.6 Prime Patterns and the Hardy–Littlewood Conjecture 1.7 Jumping Champions 1.8 The von Mangoldt Function 1.9 The Bombieri–Vinogradov Theorem 1.10 Admissible Tuples 1.11 A Brief Guide to the Literature 1.12 End Notes 2. The Sieves of Brun and Selberg 2.1 Introduction 2.2 Summary of This Chapter 2.3 Brun’s Pure Sieve 2.4 Brun’s Pure Sieve Addendum 2.5 The Selberg Sieve 2.6 Making the Constant Explicit 2.7 An Application to a Brun–Titchmarsh Inequality 2.8 Brun’s, Selberg’s and Other Sieves 2.9 A Brief Reader’s Guide to Sieve Theory 2.10 End Note: Twin Almost Primes and the Sieve Parity Problem 3. Early Work 3.1 Introduction 3.2 Chapter Summary 3.3 Erd˝os and the First Unconditional Step 3.4 The Beautiful Method of Bombieri and Davenport 3.5 Maier’s Matrix Method 3.6 End Notes 4. The Breakthrough of Goldston, Motohashi, Pintz and Yildirim 4.1 Introduction 4.2 Outline of the GPY Method 4.3 Definitions and Summary 4.4 General Preliminary Results 4.5 Special Preliminary Results 4.6 The Essential Theorem of Gallagher 4.7 The Main GPY Theorem 4.8 The Simplified Proof 4.9 GPY’s Conditional Bounded Gaps Theorem 4.10 End Notes 5. The Astounding Result of Yitang Zhang 5.1 Introduction 5.2 Summary of Zhang’s Method 5.3 Notation 5.4 Chapter Summary 5.5 Variations on the Bombieri–Vinogradov Estimates 5.6 Preliminary Lemmas 5.7 Upper Bound for the Sum S1 5.8 Lower Bound for the Sum S2 5.9 Zhang’s Prime Gap Result 5.10 End Notes 6. Maynard’s Radical Simplification 6.1 Introduction 6.2 Definitions 6.3 Chapter Summary 6.4 Selberg’s Sieve Lemmas 6.5 Other Preliminary Lemmas 6.6 Fundamental Lemmas 6.7 Integration Formulas 6.8 Maynard’s Algorithm 6.9 Main Theorems 6.10 End Notes 7. Polymath’s Refinements of Maynard’s Results 7.1 Introduction 7.2 Definitions 7.3 Chapter Summary 7.4 Preliminary Results 7.5 Polymath’s Algorithm for Mk 7.6 Limits to These Techniques: Upper Bound for Mk 7.7 Bogaert’s Krylov Basis Method 7.8 Bogaert’s Algorithm 7.9 How the Gap Bound pn+1 − pn ≤ 246 Is Derived 7.10 Limits to This Approach for Mk, 7.11 End Notes 8. Variations on Bombieri–Vinogradov 8.1 Introduction 8.2 Special Notations and Definitions 8.3 Chapter Summary 8.4 Preliminary Results 8.5 Multiple Dense Divisibility 8.6 Improving Zhang 8.7 A Fundamental Technical Result 8.8 Using Heath-Brown’s Identity 8.9 One-Dimensional Exponential Sums 8.10 Polymath’s Type I and II Estimates 8.11 Application to Prime Gaps 8.12 End Notes 9. Further Work and the Epilogue 9.1 Introduction 9.2 Assuming Elliott–Halberstam’s Conjecture 9.3 Assuming the Generalized Elliott–Halberstam Conjecture 9.4 Gaps between Almost Primes 9.5 Affine Forms and Clusters of Primes in Intervals 9.6 Limit Points of Normalized Consecutive Prime Differences 9.7 Artin’s Primitive Root Conjecture 9.8 Consecutive Primes in AP with a Fixed Common Difference 9.9 Prime Ideals and Irreducible Polynomials 9.10 Coefficients of Modular Forms 9.11 Elliptic Curves 9.12 Epilogue Appendix A. Bessel Functions of the First Kind Appendix B. A Type of Compact Symmetric Operator Appendix C. Solving an Optimization Problem Appendix D. A Brun–Titchmarsh Inequality Appendix E. The Weil Exponential Sum Bound Appendix F. Complex Function Theory Appendix G. The Dispersion Method of Linnik Appendix H. One Thousand Admissible Tuples Appendix I. PGpack Minimanual I.1 Introduction I.2 PGpack Functions References [17] [34] [51] [71] [89] [109] [128] [148] [168] [184] [205] Index AB EFG HI JKLM NOP QR ST UVWXYZ