Equivalents of the Riemann Hypothesis

This first volume of two presents classical and modern arithmetic equivalents to the Riemann hypothesis. Accompanying software is online.

Zusammenfassung This two-volume work presents the main known equivalents to the Riemann hypothesis, perhaps the most important problem in mathematics. Volume 1 presents classical and modern arithmetic equivalents, with some analytic methods. Accompanying software is freely available online.

Autorentext

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.

Inhalt

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.