Riemann Zeros and Random Matrix Theory

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.

Moments of the logarithmic derivative of characteristic polynomials from SO(2N) and USp(2N)

We study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices. In particular, we compute the asymptotics for large matrix size, $N$, of these moments evaluated at points which are approaching 1. This follows work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith where they compute these asymptotics in the case of unitary random matrices.Comment: 43 pages. This version has an added discussion and computation of lower order terms. It also contains implemented comments and suggestions from the referee for JMP. Accepted for publication in the Journal of Mathematical Physic

Discretisation for odd quadratic twists

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix Theory and Elliptic Curve

Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral weight modular forms. Numerical evidence is presented in support of them.Comment: 28 pages, 8 figure

Asymptotics of non-integer moments of the logarithmic derivative of characteristic polynomials over SO(2N+1)SO(2N+1)

This work computes the asymptotics of the non-integer moments of the logarithmic derivative of characteristic polynomials of matrices from the $SO(2N+1)$ ensemble. It follows from work of Alvarez and Snaith who computed the asymptotics of the integer moments of the same statistic over both $SO(N)$ ensembles as well as the $USp(2N)$ ensemble

Lower order terms in the full moment conjecture for the Riemann zeta function

We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide some numerical evidence in favour of the conjecture.Comment: 37 pages, 4 figure

Mixed moments of characteristic polynomials of random unitary matrices

Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painleve differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle