Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents

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

Kevin Broughan (Author)

9781107197046, Cambridge University Press

Hardback, published 2 November 2017

336 pages, 52 b/w illus. 17 tables
24.1 x 16 x 2.3 cm, 0.65 kg

'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

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.

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.

Subject Areas: History of mathematics [PBX], Functional analysis & transforms [PBKF], Complex analysis, complex variables [PBKD], Number theory [PBH]