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

Jesse Jääsaari

Abstract

We obtain $Ω$-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 $Ω$-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 $Ω$-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.

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

Abstract

We obtain -results for linear exponential sums with rational additive twists of small prime denominators weighted by Hecke eigenvalues of Maass cusp forms for the group . In particular, our -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 -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.
Paper Structure (13 sections, 38 theorems, 260 equations)

This paper contains 13 sections, 38 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Notation
  4. Rationally additively twisted $L$-functions of $\mathrm{GL}_3$ Maass cusp forms
  5. On certain Meijer $G$-functions and their asymptotics
  6. Some known estimates for long linear exponential sums
  7. Additively twisted Voronoi identities for Riesz means: the case $a\geqslant2$
  8. Additively twisted Voronoi identities for Riesz means: the case $a=1$
  9. Pointwise $\Omega$-result for long sums on $\mathrm{GL}_3$
  10. Second moments of long sums
  11. An improved pointwise bound for long sums with rational twists
  12. Short second moment for Riesz weighted sums
  13. $\Omega$-result for short sums of Fourier coefficients

Key Result

Theorem 2.1

Let $x\in[1,\infty[$ be sufficiently large and let $k$ prime so that $k\ll x^{1/3-\delta}$ for any sufficiently small fixed $\delta>0$. Then where the maximum is taken over all reduced residue classes modulo $k$.

Theorems & Definitions (59)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • ...and 49 more