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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.
List of contents
1. Nicolas' ¿(x) < li(¿(x)) equivalence; 2. Nicolas' number of divisors function equivalence; 3. An aspect of the zeta function zero gap estimates; 4. The Rogers-Tao equivalence; 5. The Dirichlet series of Dobner; 6. An upper bound for the deBruijn-Newman constant; 7. The Pólya-Jensen equivalence; 8. Ono et al. and Jensen polynomials; 9. Gonek-Bagchi universality and Bagchi's equivalence; 10. A selection of undecidable propositions; 11. Equivalences and decidability for Riemann's zeta; A. Imports for Gonek's theorems; B. Imports for Nicolas' theorems; C. Hyperbolic polynomials; D. Absolute continuity; E. Montel and Hurwitz's theorems; F. Markov and Gronwall's inequalities; G. Characterizing Riemann's zeta function; H. Bohr's theorem; I. Zeta and L-functions; J. de Reyna's expansion for the Hardy contour; K. Stirling's approximation for the gamma function; L. Propositional calculus $\mathscr{P}_0$; M. First order predicate calculus $\mathscr{P}_1$; N. Recursive functions; O. Ordinal numbers and analysis; References; Index
About the author
Kevin Broughan is an emeritus professor at the University of Waikato, New Zealand. He cofounded and is a fellow of the New Zealand Mathematical Society and the School of Computing and Mathematical Sciences. Broughan previously authored Volumes 1 and 2 of 'Equivalents of the Riemann Hypothesis' (Cambridge 2017) and 'Bounded Gaps Between Primes' (Cambridge 2021). He also wrote the software package that is part of Dorian Goldfeld's book 'Automrphic Forms and L-Functions for the Group GL(n,R)' (Cambridge 2006).
