# The mean values of cubic L-functions over function fields

## TL;DR

This paper derives an asymptotic formula for the average values of cubic L-functions over function fields, employing advanced techniques like metaplectic Eisenstein series and analyzing different cases based on the residue of q modulo 3.

## Contribution

It provides the first asymptotic formulas for mean values of cubic L-functions over function fields in both Kummer and non-Kummer cases, using novel analytic methods.

## Key findings

- Asymptotic formulas for mean values of cubic L-functions over F_q[t]
- Explicit cancellation observed between main and dual terms in the non-Kummer case
- Application of metaplectic Eisenstein series to analyze cubic Gauss sums

## Abstract

We obtain an asymptotic formula for the mean value of L-functions associated to cubic characters over F_q[t]. We solve this problem in the non-Kummer setting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the theory of metaplectic Eisenstein series. In the non-Kummer setting we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the L-functions.