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 $H\ge T^{18/19+\epsilon }$. We improve this result by showing that the estimate holds for $H\gg T^{6/7+\epsilon }$. 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 $H=T^{\beta }$ was only known to be at least $\frac{3\beta -1}{4}$. We establish an unconditional lower bound of $\frac{7\beta -2}{8}$.