Equivalents of the Riemann Hypothesis

Volume 2, Analytic Equivalents

Kevin Broughan (University of Waikato, New Zealand)

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English

Cambridge University Press
02 November 2017

Number theory; Calculus & mathematical analysis

Series: Encyclopedia of Mathematics and its Applications

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.

By: Kevin Broughan (University of Waikato New Zealand)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 165
Dimensions: Height: 241mm, Width: 162mm, Spine: 32mm
Weight: 870g
ISBN: 9781107197121
ISBN 10: 1107197120
Series: Encyclopedia of Mathematics and its Applications
Pages: 522
Publication Date: 02 November 2017
Audience: College/higher education , A / AS level , Further / Higher Education
Format: Hardback
Publisher's Status: Active

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; References; Index.

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.

Reviews for Equivalents of the Riemann Hypothesis: Volume 2, Analytic Equivalents

'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 'This two volume catalogue of many of the various equivalents of the Riemann Hypothesis by Kevin Broughan is a valuable addition to the literature ... all in all these two volumes are a must have for anyone interested in the Riemann Hypothesis.' Steven Decke, MAA Reviews 'The two volumes are a very valuable resource and a fascinating read about a most intriguing problem.' R.S. MacKay, London Mathematical Society Newsletter