The Riemann hypothesis (RH) may be the most important outstanding problem in mathematics. This third volume on equivalents to RH comprehensively presents recent results of Nicolas, Rogers–Tao–Dobner, Polymath15, and Matiyasevich. Particularly interesting are derivations which show, assuming all zeros on the critical line are simple, that RH is decidable. Also included are classical Pólya–Jensen equivalence and related developments of Ono et al. Extensive appendices highlight key background results, most of which are proved. The book is highly accessible, with definitions repeated, proofs split logically, and graphical visuals. It is ideal for mathematicians wishing to update their knowledge, logicians, and graduate students seeking accessible number theory research problems. The three volumes can be read mostly independently. Volume 1 presents classical and modern arithmetic RH equivalents. Volume 2 covers equivalences with a strong analytic orientation. Volume 3 includes further arithmetic and analytic equivalents plus new material on RH decidability.

‘This book is the third volume of the author’s work on statements equivalent to the Riemann Hypothesis (RH). This third volume is devoted largely to recent research on some of these topics. … As with the previous volumes, this book admirably demonstrates the remarkably wide variety of mathematical subject areas that have been applied to the Riemann zeta-function. Although one could hope that one of the various statements equivalent to RH might eventually lead to a proof of the hypothesis, it seems that the majority of those who have worked on these equivalent statements have done so largely because of the intrinsic interest of the mathematics. And perhaps that is the best reason for studying the present book.’

Friedrich Steinle Source: MathSciNet

‘The first edition has for thirty years been an excellent reference and it is welcome to see an updated version and have the book once again available in print.’

Skip Garibaldi Source: MathSciNet