The Integral Moments and Ratios of Quadratic Dirichlet L-Functions over Monic Irreducible Polynomials in \mathbbF_q[T]

2018/07/17 by Andrade, Julio, Jung, Hwanyup, Shamesaldeen, Asmaa
#11M38 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.06347

Abstract

In 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 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,χ_P) where the character χ is defined by the Legendre symbol for polynomials in \mathbbF_q[T] with \mathbbF_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.

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