The second moment of cubic Dirichlet L-functions over function fields

TL;DR

This paper investigates the second moment of cubic Dirichlet L-functions at the central point over function fields, extending previous work on the first moment using analytic techniques and Gauss sum estimates.

Contribution

It provides the first asymptotic formula for the second moment of cubic Dirichlet L-functions over function fields, advancing understanding beyond the first moment.

Findings

Derived an asymptotic formula for the second moment.

Connected second moment to averages of Gauss sums.

Extended prior first-moment results to second moments.

Abstract

In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result extends prior work of David, Florea and Lalin, who obtained an asymptotic formula for the first moment. Our approach relies on analytic techniques (Perron's formula, approximate functional equation, etc), adapted to the function field context. A key step in the construction is to relate second moment to certain averages of Gauss sums, which are estimated in loc. cit. using results of Kubota and Hoffstein.