Mean values with cubic characters

TL;DR

This paper studies mean value problems for order three primitive Dirichlet characters, deriving an asymptotic formula for their L-functions' central values and establishing a large-sieve result for these characters.

Contribution

It provides the first asymptotic formula for the first moment of central L-values for order three primitive Dirichlet characters with a power-saving error term.

Findings

Asymptotic formula for the first moment of central L-values

Large-sieve type result for order three and six Dirichlet characters

Power saving in the error term of the asymptotic formula

Abstract

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters.