Bounds for moments of cubic and quartic Dirichlet $L$-functions

TL;DR

This paper derives sharp bounds for the moments of central values of primitive cubic and quartic Dirichlet L-functions, advancing understanding of their size and non-vanishing properties under various hypotheses.

Contribution

It provides unconditional lower bounds for cubic L-functions and bounds under Lindelöf hypothesis for quartic L-functions, along with bounds under GRH for both.

Findings

Unconditional sharp lower bounds for cubic L-functions moments.

Bounds under Lindelöf hypothesis for quartic L-functions.

Quantitative non-vanishing results for L-values.

Abstract

We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally for the cubic case and under the Lindel\"of hypothesis for the quartic case. We also establish sharp lower bounds for all real $0 \leq k<1/2$ and sharp upper bounds for all real $k \geq 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.