Equivalents of the Riemann Hypothesis
Volume 3. Further Steps towards Resolving the Riemann Hypothesis
$195.00 USD
Part of Encyclopedia of Mathematics and its Applications
- Author: Kevin Broughan, University of Waikato, New Zealand
- Date Published: September 2023
- availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
- format: Adobe eBook Reader
- isbn: 9781009384773
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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.
Read more- Builds on Volumes 1 and 2 with new arithmetic and analytic equivalences from the work of Dobner, Nicolas, Ono et al., Polymath15, and Rogers and Tao
- Gives an accessible exploration of decidability to consider and resolve the issue of whether RH is decidable
- Offers students and researchers easy access to research problems, methods, and results
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- Date Published: September 2023
- format: Adobe eBook Reader
- isbn: 9781009384773
- availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
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
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