# The fourth power mean of Dirichlet $L$-functions in $\mathbb{F}_q [T]$

## TL;DR

This paper investigates the moments of Dirichlet L-functions over function fields, providing exact and asymptotic formulas for second and fourth moments respectively, extending prior work to more general moduli.

## Contribution

It derives new formulas for the second and fourth moments of L-functions in the function field setting, including an exact formula for square-full moduli and an asymptotic for general moduli.

## Key findings

- Exact second moment formula for square-full R
- Asymptotic fourth moment formula for any R
- Extension of previous prime modulus results

## Abstract

We prove results on moments of $L$-functions in the function field setting, where the moment averages are taken over primitive characters of modulus $R$, where $R$ is a polynomial in $\mathbb{F}_q [T]$. We consider the behaviour as $\textrm{deg} R \rightarrow \infty$ and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that $R$ is square-full, and an asymptotic formula for the fourth moment for any $R$. The fourth moment result is a function field analogue of Heath-Brown's result in the number field setting, which was subsequently improved by Soundararajan. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where $R$ is prime.