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

A mathematical record of bounded prime gaps breakthroughs, from Erd?s to Polymath8, with linked computer programs and complete appendices.

Kevin Broughan (Author)

9781108836746, Cambridge University Press

Hardback, published 25 February 2021

590 pages
25 x 17.4 x 3.5 cm, 1.14 kg

'a wonderful tale of how two lesser-known mathematicians worked extremely hard to solve an intriguing, long-standing open problem that so many leading experts could not.' Sam Chow, London Mathematical Society

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.

1. Introduction
2. The sieves of Brun and Selberg
3. Early work
4. The breakthrough of Goldston, Motohashi, Pintz, and Yildirim
5. The astounding result of Yitang Zhang
6. Maynard's radical simplification
7. Polymath's refinements of Maynard's results
8. Variations on Bombieri–Vinogradov
9. Further work and the 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 mini-manual
References
Index.

Subject Areas: History of mathematics [PBX], Number theory [PBH]