# The Fourth Moment of Derivatives of Dirichlet $L$-functions in Function Fields

## TL;DR

This paper derives the asymptotic main terms for the first, second, and mixed fourth moments of derivatives of Dirichlet L-functions over function fields, extending previous work on moments without derivatives and connecting to conjectures in number fields.

## Contribution

It provides the first asymptotic formulas for moments of derivatives of L-functions in the function field setting, including mixed moments, expanding the understanding of their distribution.

## Key findings

- Asymptotic main terms for first, second, and mixed fourth moments obtained.
- Extends previous results on moments without derivatives to derivatives.
- Supports conjectures relating to moments of derivatives in number fields.

## Abstract

We obtain the asymptotic main term of moments of arbitrary derivatives of $L$-functions in the function field setting. Specifically, the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $Q \in \mathbb{F}_q [t]$, and the asymptotic limit is as $\mathrm{deg} \, Q \longrightarrow \infty$. This extends the work of Tamam who obtained the asymptotic main term of low moments of $L$-functions, without derivatives, in the function field setting. It also expands on the work of Conrey, Rubinstein, and Snaith who cojectured, using random matrix theory, the asymptotic main term of any even moment of the derivative of the Riemann zeta-function in the number field setting.