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
'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
'All in all these books serve as a good introduction to a wide range of mathematics related to the Riemann Hypothesis and make for a valuable contribution to the literature. They are truly encyclopedic and I am sure will entice many a reader to consult some literature quoted and who knows, eventually make an own contribution to the area.' Pieter Moree, Nieuw Archief voor Wiskunde
'This book may serve as reference for the Riemann hypothesis and its equivalent formulations or as an inspiration for everyone interested in number theory. It is written in a very readable style and for most parts only assumes basic knowledge from (complex analysis). Thus it may also serve as a (somewhat specific) introduction to analytic number theory.' J. Mahnkopf, Encyclopedia of Mathematics and its Applications
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Product details
- Date Published: November 2017
- format: Hardback
- isbn: 9781107197121
- length: 522 pages
- dimensions: 241 x 162 x 32 mm
- weight: 0.87kg
- availability: Available
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
References
Index.
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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.