Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 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 Maynards 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 Minimanual
- References
- Index
5 - The Astounding Result of Yitang Zhang
Published online by Cambridge University Press: 10 September 2021
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 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 Maynards 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 Minimanual
- References
- Index
Summary
How Yitang Zhang arrived at his breakthrough result showing bounded gaps between primes unconditionally is retold in part in this chapter. GPY (Chapter 4) showed that any advance of the parameter in EH beyond one half was sufficient. In making his wonderful breakthrough, Zhang showedthat the full strength of EH beyond one half was not needed. Instead of summing over all moduli one could restrict their values to being smooth integers. In a tour de force of advanced methods, he showed that this restricted extension was unconditionally true for an explicit value of the parameter. No one before Zhang believed it to be possible to carry this through. Zhang used the methods of others, but significantly extended them to obtain sufficient flexibility to attain bounded gaps. The detail of the final part of Zhang’s argument is set out in this chapter, showing how he derived the bound. He chose explicit constants ending up with a gap size of 70 million. The ideas and methods used by Zhang come from Bombieri, Deligne, Deshouillers, Fouvry, Friedlander, Goldston, Heath-Brown, Iwaniec, Kuznetsiv, Motohashi, Pintz, Vinogradov, Yildirim and Weil and there are references given for much of this work.
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- Bounded Gaps Between PrimesThe Epic Breakthroughs of the Early Twenty-First Century, pp. 184 - 218Publisher: Cambridge University PressPrint publication year: 2021