The integral moments and ratios of quadratic Dirichlet L-functions over monic irreducible polynomials in $$\mathbb {F}_{q}[T]$$

Abstract

AbstractIn this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of L-functions. We also adapt to the function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, the focus of this paper is on the family of quadratic Dirichlet L-functions $$L(s,\chi _{P})$$ L ( s , χ P ) where the character $$\chi $$ χ is defined by the Legendre symbol for polynomials in $$\mathbb {F}_{q}[T]$$ F q [ T ] with $$\mathbb {F}_{q}$$ F q a finite field of odd cardinality, and the averages are taken over all monic and irreducible polynomials P of a given odd degree. As an application, we also compute the formula for the one-level density for the zeros of these L-functions.