Nina C Snaith

Integral moments of L -functions

We give a new heuristic for all of the main terms in the integral moments of various families of primitive

Random matrices and L-functions

L

Autocorrelation of random matrix polynomials

-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical

Developments in random matrix theory

L

Recent perspectives in random matrix theory and number theory

-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

Averages of ratios of characteristic polynomials for the compact classical groups

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.

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

Abstract: We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions.

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

In this introduction to the Journal of Physics A special issue on random matrix theory, we give a review of the main historical developments in random matrix theory. A short summary of the papers that appear in this special issue is also given.

A random matrix model for elliptic curve L-functions of finite conductor

1. Introduction 2. Prime number theory and the Riemann zeta-function 3. Notes on pair correlation of zeros and prime numbers 4. Notes on eigenvalue distributions for the classical compact groups 5. Compound nucleus resonances, random matrices and quantum chaos 6. Families of L-functions and 1-level densities 7. Basic analytic number theory 8. Applications of mean value theorems to the theory of the Riemann zeta function 9. L-functions and the characteristic polynomials of random matrices 10. Mock gaussian behaviour 11. Some specimens of L-functions 12. Computational methods and experiments in analytic number theory.

The lowest eigenvalue of Jacobi random matrix ensembles and Painleve VI

Averages of ratios of characteristic polynomials for the compact classical groups are evaluated in terms of determinants whose dimensions are independent of the matrix rank. These formulas are shown to be equivalent to expressions for the same averages obtained in a previous study, which was motivated by applications to analytic number theory. Our approach uses classical methods of random matrix theory, in particular determinants and orthogonal polynomials, and can be considered more elementary than the method of Howe pairs used in the previous study.