Equivalents of the Riemann Hypothesis

Kevin Broughan

doi:10.1017/9781009384780

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Ibrahim, Moustafa 2025. On the emergence of the Quanta Prime sequence. Arab Journal of Basic and Applied Sciences, Vol. 32, Issue. 1, p. 1.

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Volume 3: Further Steps towards Resolving the Riemann Hypothesis
Kevin Broughan, University of Waikato, New Zealand

Publisher:: Cambridge University Press
Online publication date:: October 2023
Print publication year:: 2023
Online ISBN:: 9781009384780
DOI:: https://doi.org/10.1017/9781009384780

Subjects:: Mathematics (general), Mathematics, Logic, Categories and Sets, Number Theory
Series:: Encyclopedia of Mathematics and its Applications (187)

195.00 (USD)

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

Kevin Broughan
Online ISBN: 9781009384780

Book DOI: https://doi.org/10.1017/9781009384780
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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.

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Equivalents of the Riemann Hypothesis

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Contents

Frontmatter
pp i-iv

Dedication
pp v-v

Epigraph
pp vi-vi

Contents
pp vii-xii

Preface
pp xiii-xx

1 - Nicolas’ π(x) <li(θ(x)) Equivalence
pp 1-34

2 - Nicolas’ Number of Divisors Function Equivalence
pp 35-87

3 - An Aspect of the Zeta Function Zero Gap Estimates
pp 88-108

4 - The Rogers–Tao Equivalence
pp 109-174

5 - The Dirichlet Series of Dobner
pp 175-207

6 - An Upper Bound for the de Bruijn–Newman Constant
pp 208-288

7 - The Pólya–Jensen Equivalence
pp 289-306

8 - Ono et al. and Jensen Polynomials
pp 307-336

9 - Gonek–Bagchi Universality and Bagchi’s Equivalence
pp 337-387

10 - A Selection of Undecidable Propositions
pp 388-423

11 - Equivalences and Decidability for Riemann’s Zeta
pp 424-470

Appendix A - Imports for Gonek’s Theorems
pp 471-489

Appendix B - Imports for Nicolas’ Theorems
pp 490-494

Appendix C - Hyperbolic Polynomials
pp 495-515

Appendix D - Absolute Continuity
pp 516-525

Appendix E - Montel’s and Hurwitz’s Theorems
pp 526-529

Appendix F - Markov’s and Gronwall’s Inequalities
pp 530-531

Appendix G - Characterizing Riemann’s Zeta Function
pp 532-540

Appendix H - Bohr’s Theorem
pp 541-543

Appendix I - Zeta and L–Functions
pp 544-552

Appendix J - de Reyna’s Expansion for the Hardy Contour
pp 553-564

Appendix K - Stirling’s Approximation for the Gamma Function
pp 565-566

Appendix L - Propositional Calculus P0
pp 567-578

Appendix M - First Order Predicate Calculus P1
pp 579-625

Appendix N - Recursive Functions
pp 626-648

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