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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.Read more
- 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
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 ReviewsSee more 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
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- Date Published: December 2017
- format: Hardback
- isbn: 9781107197121
- length: 522 pages
- dimensions: 241 x 162 x 32 mm
- weight: 0.87kg
- availability: Available
Table of Contents
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
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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