Equivalents of the Riemann Hypothesis: Volume 1, Arithmetic Equivalents (Encyclopedia of Mathematics and its Applications, Series Number 164)

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. "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 proof."--Back cover 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. This two-volume work covers the main known equivalents to the Riemann hypothesis. Classical and modern arithmetic statements equivalent to the Riemann hypothesis are elucidated, proved and applied using analytic and computational methods. Each volume can be read independently. Volume 1. Arithmetic equivalents -- volume 2. Analytic equivalents. Kevin Broughan, University of Waikato, New Zealand. Includes bibliographical references and indexes.