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

Jiseong Kim

Published 2021-10-13, updated 2022-05-18Version 3

By assuming the Vinogradov-Korobov type zero-free regions and the generalized Ramanujan-Petersson conjecture, we give some nontrivial upper bounds of almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for $SL(n,\mathbb{Z})$. As an application, we obtain the nontrivial upper bounds for the averages of shifted sums for $SL(n,\mathbb{Z}) \times SL(n,\mathbb{Z}).$ We also give some results on sign changes by assuming some zero-free regions and the generalized Ramanujan-Petersson conjecture.