On squares of Fourier coefficients twist exponential functions with applications

Abstract

Let f be a Hecke–Maass cusp form of weight zero for [Formula: see text] and [Formula: see text] be the nth Fourier coefficient. For almost all [Formula: see text] we have [Formula: see text] which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than [Formula: see text] For all [Formula: see text] of type [Formula: see text] we also show that [Formula: see text] where [Formula: see text] This result relies heavily on a generalized double sum (see Theorem 1.3).