The Integral Moments and Ratios of Quadratic Dirichlet $L$-Functions over Monic Irreducible Polynomials in $\mathbb{F}_{q}[T]$

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, χP ) where the character χ is defined by the Legendre symbol for polynomials in Fq[T ] with Fq 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.

Moments of Dirichlet L–functions with prime conductors over function fields

“…Here we check that the asymptotic formula we obtain in Theorem 1.1 agrees with the conjecture in [AJS18]. From [AJS18] we have that the term involving g 2 in the asymptotic formula should be equal to 3(2g) 2 24 A( 1 2 ; 0, 0) + 6(2g) 2 24 A( 1 2 ; 0, 0) + 3(2g) 2 24 log q A 1 ( 1 2 ; 0, 0) + A 2 ( 1 2 ; 0, 0) , (8.1)…”

“…Similar to the recipe developed by Conrey, Farmer, Keating, Rubinstein and Snaith [CFK + 05], Andrade, Jung and Shamesaldeen [AJS18] conjectured asymptotic formulas for the integral moments of L(1/2, χ P ). Specifically, the conjecture is that 1…”

Hybrid Euler-Hadamard Product for Dirichlet $L$-functions with Prime conductors over Function Fields

“…Andrade and Keating [1] established an asymptotic formulas for I k (g) when k = 1, 2 and in a recent paper Bui and Florea [9] improved the error term in the formula for I 2 (g) obtaining an extra lower order term. For other values of k, following the recipe to the function field setting [3], Andrade, Jung and Shamesaldeen [5] proposed a general formula for the integral moments of quadratic Dirichlet L-functions associated to χ P over function fields.…”