On some estimates involving Fourier coefficients of Maass cusp forms

Abstract

Let [Formula: see text] be a Hecke–Maass cusp form for [Formula: see text] with Laplace eigenvalue [Formula: see text] and let [Formula: see text] be its [Formula: see text]th normalized Fourier coefficient. It is proved that, uniformly in [Formula: see text], [Formula: see text] where the implied constant depends only on [Formula: see text]. We also consider the summation function of [Formula: see text] and under the Ramanujan conjecture we are able to prove [Formula: see text] with the implied constant depending only on [Formula: see text].