Additive twists of Fourier coefficients of modular forms

Abstract

We study sums of the form ∑ n ⩽ N a ( n ) e 2 π i α n , where α is any real number and the a ( n ) are the Fourier coefficients of either a holomorphic cusp form, a Maass cusp form, or the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform in both α and the form itself. We also improve a bound on a sum of the form ∑ n ⩽ N a ( n ) e 2 π i ( α n + β n θ ) , where the a ( n ) are the Fourier coefficients of a holomorphic cusp form, α and β are any real numbers, and 0 ⩽ θ < 1 . This last bound is uniform in α , but not with respect to the form.