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

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 $k\ge 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0\le k<1/2$ and sharp upper bounds for all real $k\ge 0$ 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.

24 pages

References in corpus (4)

• Lower bounds for moments of zeta and $L$-functions revisited
• Moments of central values of quartic Dirichlet $L$-functions
• Sharp upper bounds for moments of quadratic Dirichlet $L$-functions
• Bounds for moments of Dirichlet $L$-functions to a fixed modulus

Cited by in corpus (1)

• Moments of quadratic Dirichlet $L$-functions over Function Fields