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

TL;DR

This paper derives new bounds for short sums and averages of Fourier coefficients of Hecke-Maass cusp forms on SL(n,Z), assuming certain zero-free regions and conjectures, and discusses sign changes of these coefficients.

Contribution

It introduces conditional bounds for sums of Hecke eigenvalues and explores sign change phenomena under specific hypotheses.

Findings

Established upper bounds for short sums of Fourier coefficients.

Obtained bounds for averages of shifted sums of Hecke eigenvalues.

Presented a conditional result on sign changes of coefficients.

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.