## Ph.D. Thesis

**Random Matrix Theory and $L$-functions in Function Fields**

PhD Thesis, University of Bristol, 2012

Advisor: J.P. Keating and N.C. Snaith

**Abstract:**It is an important problem in analytic number theory to estimate mean values of the Riemann zeta-function and other $L$-functions. The study of moments of $L$-functions has some important applications, such as to give information about the maximal order of the Riemann zeta-function on the critical line, Lindelöf Hypothesis for $L$-functions and non-vanishing results. Furthermore, according to the Katz-Sarnak philosophy it is believed that the understanding of mean values of different families of $L$-functions may reveal the symmetry of such families.

The analogy between characteristic polynomials of random matrices and $L$-functions was first studied by Keating and Snaith. For example, they were able to conjecture asymptotic formulae for the moments of $L$-functions in different families. The purpose of this thesis is to study moments of $L$-functions over function fields, since in this case, the $L$-functions satisfy a Riemann Hypothesis and one may give a spectral interpretation for such $L$-functions as the characteristic polynomial of a unitary matrix. Thus, we expect that the analogy between characteristic polynomials and $L$-functions can be further understood in this scenario.

In this thesis, we study power moments of a family of $L$-functions associated with hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_{q}$ in the limit as $g\rightarrow\infty$, which is the opposite limit considered by the programme of Katz and Sarnak. Specifically, we compute some average value theorems of $L$-functions of curves and we extend to the function field setting the heuristic for integral moments and ratios of $L$-functions previously developed by Conrey et.al. for the number field case.