Volume 1. Analytic Equivalents: 1. Introduction; 2. The Riemann Zeta function; 3. Estimates; 4. Classical equivalences; 5. Euler's Totient function; 6. A variety of abundant numbers; 7. Robin's theorem; 8. Numbers which do not satisfy Robin's inequality; 9. Left, right and extremely abundant numbers; 10. Other equivalents to the Riemann hypothesis; Appendix A. Tables; Appendix B. RHpack mini-manual; Bibliography; Index. Volume 2. Arithmetic Equivalents: 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.
This two-volume work presents the main known equivalents to the Riemann hypothesis. Each volume can be read independently.
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.
'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
'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
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