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Bounded Gaps Between Primes

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

c.$49.99 ( )

  • Publication planned for: March 2021
  • availability: Not yet published - available from March 2021
  • format: Paperback
  • isbn: 9781108799201

c.$ 49.99 ( )
Paperback

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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
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    Product details

    • Publication planned for: March 2021
    • format: Paperback
    • isbn: 9781108799201
    • dimensions: 244 x 170 mm
    • availability: Not yet published - available from March 2021
  • 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.

  • Author

    Kevin Broughan, University of Waikato, New Zealand
    Kevin Broughan is Emeritus Professor at the University of Waikato, New Zealand. He co-founded and is a Fellow of the New Zealand Mathematical Society. Broughan brings a unique set of knowledge and skills to this project, including number theory, analysis, topology, dynamical systems and computational mathematics. He previously authored the two-volume work Equivalents of the Riemann Hypothesis (Cambridge, 2017) and wrote a software package which is part of Goldfeld's Automorphic Forms and L-Functions for the Group GL(n,R) (Cambridge, 2006).

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