The mean values of cubic L-functions over function fields

Abstract

We obtain an asymptotic formula for the mean value of $L$-functions associated to cubic characters over ${\mathbb{𝔽}}_q[T]$. We solve this problem in the non-Kummer setting when $q\equiv 2(\text{mod }3)$ and in the Kummer setting when $q\equiv 1(\text{mod }3)$. In the Kummer setting, the mean value over the complete family of cubic characters was never addressed in the literature (over number fields or function fields). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the pioneer work of Kubota. In the non-Kummer setting, we display some explicit (and unexpected) cancellation between the main term and the dual term coming from the approximate functional equation of the $L$-functions. Exhibiting the cancellation involves evaluating sums of residues of a variant of the generating series of cubic Gauss sums.