On some estimates involving Fourier coefficients of Maass cusp forms

TL;DR

This paper establishes bounds on sums involving Fourier coefficients of Maass cusp forms, providing uniform estimates and conditional results under the Ramanujan conjecture, advancing understanding of their oscillatory behavior.

Contribution

It proves new uniform bounds for exponential sums of Fourier coefficients of Maass cusp forms and conditional estimates assuming the Ramanujan conjecture.

Findings

Bound for quadratic exponential sums of Fourier coefficients: $X^{7/8+ ext{ε}}$

Conditional sum estimate under Ramanujan conjecture: $X^{1/3+ ext{ε}}$

Results depend only on ε and eigenvalue $ u_f( riangle)$

Abstract

Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $\lambda_f(\Delta)=1/4+\mu^2$ and let $\lambda_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $\alpha, \beta \in \mathbb{R}$, $$ \sum_{n \leq X}\lambda_f(n)e\left(\alpha n^2+\beta n\right) \ll X^{7/8+\varepsilon}\lambda_f(\Delta)^{1/2+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$. We also consider the summation function of $\lambda_f(n)$ and under the Ramanujan conjecture we are able to prove $$ \sum_{n \leq X}\lambda_f(n)\ll X^{1/3+\varepsilon}\lambda_f(\Delta)^{4/9+\varepsilon} $$ with the implied constant depending only on $\varepsilon$.