Resonances and $\Omega$-results for Exponential Sums Related to Maass Forms for $\mathrm{SL}(n,\mathbb Z)$

TL;DR

This paper derives asymptotics for exponential sums involving Fourier coefficients of Maass forms for SL(n,Z), establishing resonances and Omega-results that advance understanding of their oscillatory behavior.

Contribution

It introduces new asymptotic formulas for integrals in the GL(n) Voronoi summation and proves Omega-results for short sums of Fourier coefficients, extending previous work.

Findings

Resonances identified for short exponential sums of Maass form coefficients.

Asymptotic formulas derived for integrals in the GL(n) Voronoi formula.

Omega-results established for short sums, indicating oscillatory lower bounds.

Abstract

We obtain resonances for short exponential sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(n,\mathbb Z)$. This involves deriving asymptotics for the integrals appearing in the $\mathrm{GL}(n)$ Voronoi summation formula. As an application, we also prove an $\Omega$-result for short sums of Fourier coefficients.