Bounded gaps between primes : the epic breakthroughs of the early twenty-first century
著者
書誌事項
Bounded gaps between primes : the epic breakthroughs of the early twenty-first century
Cambridge University Press, 2021
- : hard
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注記
Includes bibliographical references (p. 555-566) and index
内容説明・目次
内容説明
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.
目次
- 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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