Moments and Non-Vanishing of Maass Form Symmetric Square L-Functions in Short Intervals

Abstract: Recently, Li obtained a mean Lindelöf estimate for the cubic moment of the central values of Maass form symmetric square $L$-functions over short intervals $(T-H, T+H)$ of length $H \ge T<sup>{18/19+ε}$. We improve this result by showing that the estimate holds for $H \gg T<sup>{6/7+ε}$. The key ingredient in our proof is a new asymptotic formula for the second twisted moment of Maass symmetric square $L$-functions. Based on this formula, we also improve the lower bound for the proportion of non-vanishing central $L$-values in short intervals. Previously, even under the assumption of the Lindelöf hypothesis for Dirichlet $L$-functions, the proportion of non-vanishing values in intervals of length $H = T<sup>β$ was only known to be at least $\frac{3β-1}{4}$. We establish an unconditional lower bound of $\frac{7β-2}{8}$.