Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

Abstract

By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for $SL(n,\mathbb{Z})$. As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for $SL(n,\mathbb{Z})$. Furthermore, we present a conditional result regarding sign changes of these coefficients.