Topics on L-functions in Function fields

Abstract

In this thesis, we extend the work of Andrade and Keating [8] and we consider the family of quadratic Dirichlet L-functions associated to monic and irreducible polynomials over a finite field. We then proceed to study the moments conjecture and the ratios conjecture for this family of L-functions. We also compute the lower order terms for the n-correlation of zeros of the chosen family of L-functions. These calculations follow the work of Mason and Snaith [83] carried for families of L-functions for the classical case. We establish for the function field setting the analogue of a result first proved by Burr [21] in the number field case. A novelty of this thesis is that we are able to extend Burr’s result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions d_k(f) with an error term that improves the one given by Burr in the number field case.