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

TL;DR

This paper extends heuristic models for moments and ratios of quadratic Dirichlet L-functions to the function field setting, focusing on polynomials over finite fields, and computes the one-level density of their zeros.

Contribution

It adapts and extends conjectural heuristics for moments and ratios of L-functions to the function field context, providing new predictions and formulas.

Findings

Heuristic models for moments and ratios are successfully adapted to function fields.

Derived a formula for the one-level density of zeros of quadratic Dirichlet L-functions.

Provides a framework for future rigorous verification in the function field setting.

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 $L(s,\chi_{P})$ where the character $\chi$ is defined by the Legendre symbol for polynomials in $\mathbb{F}_{q}[T]$ with $\mathbb{F}_{q}$ 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.