$\Omega$-Results for Exponential Sums Related to Maass Cusp Forms for $\mathrm{SL}_3(\mathbb Z)$

TL;DR

This paper establishes lower bounds and matches conjectural bounds for exponential sums involving Maass cusp forms on SL_3(Z), advancing understanding of their size and behavior in number theory.

Contribution

It provides new Omega-results for exponential sums with small prime denominators, matching conjectural bounds, and improves bounds for sums over short segments.

Findings

Omega-results match conjectural bounds for small denominators

Lower bounds for mean squares of exponential sums

Improved upper bounds in certain parameter ranges

Abstract

We obtain $\Omega$-results for linear exponential sums with rational additive twists of small prime denominators weighted by Hecke eigenvalues of Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. In particular, our $\Omega$-results match the expected conjectural upper bounds when the denominator of the twist is sufficiently small compared to the length of the sum. Non-trivial $\Omega$-results for sums over short segments are also obtained. Along the way we produce lower bounds for mean squares of the exponential sums in question and also improve the best known upper bound for these sums in some ranges of parameters.