First moment of central values of some primitive Dirichlet $L$-functions with fixed order characters

TL;DR

This paper asymptotically evaluates the average size of central values of primitive Dirichlet L-functions with fixed order characters and establishes non-vanishing results, advancing understanding of their distribution.

Contribution

It introduces the use of double Dirichlet series to evaluate moments of primitive Dirichlet L-functions with fixed order characters, providing new asymptotic formulas and non-vanishing results.

Findings

Asymptotic formulas for the first moment of central L-values

Quantitative non-vanishing results for these L-values

Application of double Dirichlet series method

Abstract

We evaluate asymptotically the smoothed first moment of central values of families of primitive cubic, quartic and sextic Dirichlet $L$-functions, using the method of double Dirichlet series. Quantitative non-vanishing result for these $L$-values are also proved.