On some estimates involving Fourier coefficients of Maass cusp forms

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 $\varepsilon$.