On some estimates involving Fourier coefficients of Maass cusp forms

Abstract

Let $f$ be a Hecke-Maass cusp form for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ with Laplace eigenvalue $\hat{I}{»}_f(\hat{I}”)=1/4+\hat{I}{¼}^2$ and let $\hat{I}{»}_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $\hat{I}\pm ,{\hat{I}}^2\in \mathbb{R}$, $$\sum _{n\le X}\hat{I}{»}_f(n)e\left(\hat{I}\pm n^2+{\hat{I}}^{2}n\right)\ll X^{7/8+\epsilon }\hat{I}{»}_f(\hat{I}”{)}^{1/2+\epsilon },$$ where the implied constant depends only on $\epsilon$. We also consider the summation function of $\hat{I}{»}_f(n)$ and under the Ramanujan conjecture we are able to prove $$\sum _{n\le X}\hat{I}{»}_f(n)\ll X^{1/3+\epsilon }\hat{I}{»}_f(\hat{I}”{)}^{4/9+\epsilon }$$ with the implied constant depending only on $\epsilon$.