Bounds for moments of cubic and quartic Dirichlet L-functions

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≥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≤k<1/2 and sharp upper bounds for all real k≥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.