Riemann Zeros and Random Matrix Theory

Abstract

In the past dozen years random matrix theory has become a useful tool for conjecturing answers to old and important questions in number theory. It was through the Riemann zeta function that the connection with random matrix theory was first made in the 1970s, and although there has also been much recent work concerning other varieties of L-functions, this article will concentrate on the zeta function as the simplest example illustrating the role of random matrix theory.

Contrasting SYK-like models

“…Furthermore, these models are a suggestion that there do exist 0+1 dimensional quantum mechanical models that could capture quantum gravity in AdS 2 . There exists another system which is expected to be modeled by a 0+1 dimensional quantum mechanical system that exhibits random matrix behavior and is intimately connected to quantum chaos [29,30]. This is the putative Hamiltonian whose eigenvalues are supposed to reproduce the non-trivial zeros of the Riemann zeta function.…”

A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

“…In 1973, Montgomery [38] conjectured that, assuming the Riemann hypothesis, the distances between appropriately normalised pairs of zeros of the Riemann zeta function follow a certain distribution previously shown by Dyson [15] to describe spacings between pairs of eigenvalues of unitary random matrices. Further evidence of a connection between number theory and random matrix theory was given when Keating and Snaith [32,33] used results for the characteristic polynomial of a random unitary matrix to formulate conjectures about moments of the zeta function that are supported by number-theoretic and numerical results (see, for instance, the review articles [30,45,34,31]). Since then, a number of more general results on the joint moments of the characteristic polynomial and its derivative have been proven and used to formulate conjectures about the joint moments of L-functions and their derivatives [7,8,9,10,12,23,24,46].…”

A Central Limit Theorem for the Zeroes of the Zeta Function

“…Indeed, one can already discern, by comparing Fujii's central limit theorem to the central limit theorem of Costin and Lebowitz, that the statistics of the zeros in this regime cannot be modeled too closely by a sine-kernel determinantal process. Outside of the mesoscopic regime, these statistics demonstrate an important 'resurgence phenomenon' discovered heuristically by Bogomolny and Keating, and explored in [4], [23], [25] and [31]. Zeev Rudnick pointed out to the author that he had used similar ideas with Faifman in [10] to prove a Fujii-type central limit theorem, for counting functions with a strict cutoff, in the finite field setting.…”