Weight aspect exponential sums for Fourier coefficients of cusp forms
Let \(f\) be a primitive holomorphic cusp form of positive even weight \(k\) and denote its \(n\)-th Fourier coefficient by \(\lambda_f(n)\). The paper under review concerns about a uniform bound on the exponential sum of Fourier coefficients \(\lambda_f(n)\) with the twists. More precisely, the author studies the explicit dependence for the quadratic exponential sums, as a way to explore new behavior of the form in these types of sums. He aims to obtain an estimate in terms of the weight parameter. As an application of the main theorem, he obtains the explicit dependence on the weight aspect for the quadratic exponential sums involving Fourier coefficients of cusp forms at prime arguments. The proofs are mainly based on calculations through Voronoï-type summation formula, a Wilton-type bound, Bessel functions and the circle method.
scientific article; zbMATH DE number 7603028
scientific article