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
9 - Further Work and the Epilogue
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
Further results are flowing like a stream. By early 2018 the principal works which have formed the core of this book had collectively received over 210 citations registered on MathSciNet. In this chapter we summarize some of these results, depending also on the list of Andrew Granville. The material covered is but an indication of the fundamental nature of the breakthroughs which have been described, and much more is expected in the future. This chapter gives a description of the results but without proofs. The topics include bounded gaps for primes described by affine forms, clusters of primes in intervals, gaps between almost primes, clusters of primes in intervals, the set of limit points of the sequence of normalized consecutive prime differences, Artin’s primitive root conjecture, arithmetic progressions of primes with a fixed common difference, prime ideals inthe ring of integers of number fields, irreducible polynomials, the coefficients of a class of modular forms including Ramanujan’s tau-function form, quadratic twistsof a class of elliptic curves including the congruent number elliptic curve, and results obtained assuming the Elliott–Halberstam and generalized Elliott–Halberstam conjectures. The results often take the form of establishing bounded gaps for primes of a particular form.
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- Bounded Gaps Between PrimesThe Epic Breakthroughs of the Early Twenty-First Century, pp. 451 - 464Publisher: Cambridge University PressPrint publication year: 2021