scholarly journals Pair correlation and twin primes revisited

2016 ◽  
Vol 472 (2194) ◽  
pp. 20160548 ◽  
Author(s):  
Brian Conrey ◽  
Jonathan P. Keating
Keyword(s):  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Arithmetic Function ◽  
Pair Correlation ◽  
Twin Primes ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We establish a connection between the conjectural two-over-two ratios formula for the Riemann zeta-function and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe.

The Riemann zeta-function and moment conjectures from Random Matrix Theory

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
Jörn Steuding
Keyword(s):  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Limited Range ◽  
Theory Model ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Moments

AbstractOn the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line, $$ \frac{1} {T}\int\limits_0^T {|\zeta (\tfrac{1} {2} + it)|^{2k} dt} and \frac{1} {{N(T)}}\sum\limits_{0 < \gamma \leqslant {\rm T}} {|\zeta (\tfrac{1} {2} + i(\gamma + \tfrac{\alpha } {L}))|^{2k} } $$, by Conrey, Keating et al. and Hughes, respectively. These conjectures are known to be true only for a few values of k and, even under assumption of the Riemann hypothesis, estimates of the expected order of magnitude are only proved for a limited range of k. We put the discrete moment for k = 1, 2 in relation with the corresponding continuous moment for the derivative of Hardy’s Z-function. This leads to upper bounds for the discrete moments which are off the predicted order by a factor of log T.

scholarly journals Uniform asymptotics of the coefficients of unitary moment polynomials

2010 ◽  
Vol 467 (2128) ◽  
pp. 1073-1100 ◽  
Author(s):  
G. A. Hiary ◽  
M. O. Rubinstein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Numerical Data ◽  
Unitary Matrix ◽  
Uniform Asymptotics ◽  
Absolute Moment ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Keating and Snaith showed that the 2 k th absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree k 2 . In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in an explicit form. Numerical data to support these calculations are presented. Some apparent connections between the random matrix theory and the Riemann zeta function are discussed.

scholarly journals Weighted value distributions of the Riemann zeta function on the critical line

2021 ◽  
Vol 33 (3) ◽  
pp. 579-592
Author(s):  
Alessandro Fazzari
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Formula ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Matrix Theory ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove a central limit theorem for log ⁡ | ζ ⁢ ( 1 2 + i ⁢ t ) | {\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure | ζ ( m ) ⁢ ( 1 2 + i ⁢ t ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} ( k , m ∈ ℕ {k,m\in\mathbb{N}} ), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure | ζ ⁢ ( 1 2 + i ⁢ t + i ⁢ α ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta(\frac{1}{2}+it+i\alpha)\rvert^{2k}\,dt} , with α ∈ ( - 1 , 1 ) {\alpha\in(-1,1)} . Finally, we prove unconditionally the analogue result in the random matrix theory context.

Number theory

2018 ◽  
Author(s):  
Thomas Spencer
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Extreme Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Eigenvalues Of Random Matrices ◽  
Areas Of Interest

This article examines some of the connections between random matrix theory (RMT) and number theory, including the modelling of the value distributions of the Riemann zeta function and other L-functions as well as the statistical distribution of their zeros. Number theory has been used in RMT to address seemingly disparate questions, such as modelling mean and extreme values of the Riemann zeta function and counting points on curves. One thing in common among the applications of RMT to number theory is the L-function. The statistics of the critical zeros of these functions are believed to be related to those of the eigenvalues of random matrices. The article first considers the truth of the generalized Riemann hypothesis before discussing the values of the Riemann zeta function, the values of L-functions, and further areas of interest with respect to the connections between RMT and number theory

scholarly journals HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS

2020 ◽  
Vol 71 (1) ◽  
pp. 257-280
Author(s):  
Jeffrey C Lagarias ◽  
Brad Rodgers
Keyword(s):  
Correlation Function ◽  
Zeta Function ◽  
Point Process ◽  
Riemann Zeta Function ◽  
Alternative Hypothesis ◽  
Pair Correlation ◽  
Local Statistics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Rule Out

Abstract The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

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