Moments of Traces of Frobenius of Higher Order Dirichlet $L$-functions over $\mathbb{F}_q[T]$

TL;DR

This paper investigates the moments of traces of Frobenius for higher order Dirichlet L-functions over finite fields, revealing their asymptotic behavior aligns with moments of unitary matrices, indicating deep connections between number theory and random matrix theory.

Contribution

It establishes the asymptotic behavior of moments of Frobenius traces for higher order Dirichlet L-functions over finite fields, linking them to random matrix theory predictions.

Findings

Moments of Frobenius traces match those of unitary matrices after normalization.

Asymptotic behavior described by a weighted group of unitary matrices.

Results extend understanding of L-functions over finite fields and their statistical properties.

Abstract

We study the moments of $\mbox{Tr}(\Theta_\chi)$ as $\chi$ runs over Dirichlet characters defined over $\mathbb{F}_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behave like the power sum moments of the group of unitary matrices multiplied by a weight function.