Bounded Gaps Between Primes
Bounded Gaps Between Primes
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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.
- Lays out a path from the twentieth century to the best results of the twenty-first century, allowing readers to dip in and out
- Gives introductory and supporting background proofs and results including Bessel functions, compact operators, complex analysis and Weil's theorem for curves
- Includes software-engineered and documented computer programs so readers can attempt to improve the best results, for example by changing the code
Reviews & endorsements
‘The author has gathered almost 100 year's worth of progress on this family of problems into one volume, and this alone will be very helpful to anyone pursuing research in the field. Recommended.’ M. Bona, Choice
‘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
Product details
- Published: April 2021
- Format: Hardback
- ISBN: 9781108836746
- Length: 590 pages
- Dimensions: 250 × 174 × 35 mm
- Weight: 1.14kg
- Availability: Available
Table of Contents
- 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.
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