scholarly journals Maxima of log-correlated fields: some recent developments*

2022 ◽  
Vol 55 (5) ◽  
pp. 053001
Author(s):  
E C Bailey ◽  
J P Keating
Keyword(s):  
Recent Progress ◽  
Symmetric Functions ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Compact Groups ◽  
Gaussian Fields ◽  
The Riemann Zeta Function ◽  
Recent Developments ◽  
Explicit Formulae ◽  
Riemann Zeta

Abstract We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.

scholarly journals Freezing transitions and extreme values: random matrix theory, and disordered landscapes

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120503 ◽  
Author(s):  
Yan V. Fyodorov ◽  
Jonathan P. Keating
Keyword(s):  
Matrix Theory ◽  
Extreme Values ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Constant Length ◽  
Energy Models ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Transition Scenario ◽  
Unitary Ensemble

We argue that the freezing transition scenario , previously conjectured to occur in the statistical mechanics of 1/ f -noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p N ( θ ) of large N × N random unitary (circular unitary ensemble) matrices U N ; i.e. the extreme value statistics of p N ( θ ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μ N ( x ) of the intervals in which | p N ( θ )|> N x , x >0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ ( s ) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

scholarly journals On the vertical distribution of values of L-functions in the Selberg class

2021 ◽  
pp. 1-26
Author(s):  
Athanasios Sourmelidis ◽  
Teerapat Srichan ◽  
Jörn Steuding
Keyword(s):  
Vertical Distribution ◽  
Selberg Class ◽  
Distribution Of Zeros ◽  
Universality Theorem ◽  
Distribution Of Values ◽  
The Riemann Zeta Function ◽  
Explicit Formulae ◽  
Riemann Zeta ◽  
Discrete Universality ◽  
The Vertical Distribution

We prove explicit formulae for [Formula: see text]-points of [Formula: see text]-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function [Formula: see text] to the case of [Formula: see text]-points of the aforementioned [Formula: see text]-functions. This result implies the uniform distribution of subsequences of [Formula: see text]-points and from this a discrete universality theorem in the spirit of Voronin is derived.

Random matrices and number theory: some recent themes

2018 ◽  
Author(s):  
Jon P. Keating
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gaussian Random Fields ◽  
Extreme Value Statistics ◽  
Matrix Methods ◽  
Recent Developments ◽  
Riemann Zeta

The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

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