Equivalents of the Riemann Hypothesis: Volume 3, Further Steps towards Resolving the Riemann Hypothesis

This third volume presents further equivalents to the Riemann hypothesis and explores its decidability.

Kevin Broughan (Author)

9781009384803, Cambridge University Press

Hardback, published 12 October 2023

706 pages
24 x 16.2 x 4.8 cm, 1.27 kg

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.

1. Nicolas' π(x) < li(θ(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

Subject Areas: Number theory [PBH]