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Equivalents of the Riemann Hypothesis

Volume 2. Analytic Equivalents

$124.00 ( ) USD

Part of Encyclopedia of Mathematics and its Applications

  • Date Published: November 2017
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9781108195430

$ 124.00 USD ( )
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About the Authors
  • The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proofs.

    • Gives students and researchers easy access to methods and results
    • Fully describes approaches to the Riemann hypothesis using analytic and functional analytic methods
    • Provides reviews of modern generalisations and tailored software
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    Reviews & endorsements

    'Throughout the book careful proofs are given for all the results discussed, introducing an impressive range of mathematical tools. Indeed, the main achievement of the work is the way in which it demonstrates how all these diverse subject areas can be brought to bear on the Riemann hypothesis. The exposition is accessible to strong undergraduates, but even specialists will find material here to interest them.' D. R. Heath-Brown, Mathematical Reviews

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    Product details

    • Date Published: November 2017
    • format: Adobe eBook Reader
    • isbn: 9781108195430
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • Table of Contents

    1. Introduction
    2. Series equivalents
    3. Banach and Hilbert space methods
    4. The Riemann Xi function
    5. The de Bruijn-Newman constant
    6. Orthogonal polynomials
    7. Cyclotomic polynomials
    8. Integral equations
    9. Weil's explicit formula, inequality and conjectures
    10. Discrete measures
    11. Hermitian forms
    12. Dirichlet L-functions
    13. Smooth numbers
    14. Epilogue
    Appendix A. Convergence of series
    Appendix B. Complex function theory
    Appendix C. The Riemann-Stieltjes integral
    Appendix D. The Lebesgue integral on R
    Appendix E. Fourier transform
    Appendix F. The Laplace transform
    Appendix G. The Mellin transform
    Appendix H. The gamma function
    Appendix I. Riemann Zeta function
    Appendix J. Banach and Hilbert spaces
    Appendix K. Miscellaneous background results
    Appendix L. GRHpack mini-manual

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    Equivalents of the Riemann Hypothesis

    Kevin Broughan

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  • Author

    Kevin Broughan, University of Waikato, New Zealand
    Kevin Broughan is Emeritus Professor in the Department of Mathematics and Statistics at the University of Waikato, New Zealand. In these two volumes he has used a unique combination of mathematical knowledge and skills. Following the publication of his Columbia University thesis, he worked on problems in topology before undertaking work on symbolic computation, leading to development of the software system SENAC. This led to a symbolic-numeric dynamical systems study of the zeta function, giving new insights into its behaviour, and was accompanied by publication of the software GL(n)pack as part of D. Goldfeld's book, Automorphic Forms and L-Functions for the Group GL(n,R). Professor Broughan has published widely on problems in prime number theory. His other achievements include co-establishing the New Zealand Mathematical Society, the School of Computing and Mathematical Sciences at the University of Waikato, and the basis for New Zealand's connection to the internet.

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