Conjectures for the Integral Moments and Ratios of L-functions in Even characteristic

TL;DR

This paper extends heuristic models for the moments and ratios of L-functions to the setting of function fields over finite fields of characteristic two, providing asymptotic formulas and zero-density results.

Contribution

It adapts and applies conjectural frameworks for L-function moments and ratios to quadratic Dirichlet L-functions over function fields, deriving new asymptotic formulas.

Findings

Asymptotic formulas for moments and ratios of L-functions in function fields.

Calculation of the one-level density for zeros of these L-functions.

Extension of heuristic models to characteristic two function fields.

Abstract

In this paper, we extend to the function field setting the heuristics developed by Conrey, Farmer, Keating, Rubinstein and Snaith for the integral moments of L-functions. Also, we adapt to function field setting the heuristics first developed by Conrey, Farmer and Zirnbauer to the study of mean values of ratios of L-functions. Specifically, we obtain an asymptotic formula for the integral moments and ratios of the quadratic Dirichlet L-functions $L(s,\chi_u)$ over the rational function field $\mathbb{F}_q(T)$, when $q$ is a power of 2 and over a given family. As an application, we calculate the one-level density for the zeros of these L-functions.