Zero free region for spectral averages of Hecke–Maass L-functions

Abstract

We provide a non-vanishing region for an infinite sum of weight zero Hecke–Maass L-functions for the full modular group inside the critical strip. For given positive parameters T and \(1 \leq M \ll \frac{T}{\log T}\), T large, we also count the number of Hecke–Maass cusp forms whose L-values are non-zero at any point s in this region and whose spectral parameters \(t_j\) lie in short intervals.