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

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 \(1 \leq M \ll \frac{T}{\log T}\) $1\le M\ll \frac{T}{logT}$ , 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 \(t_j\) $t_j$ lie in short intervals.