On sums of Fourier coefficients of cusp forms twisted with additive characters

Abstract

When $a_n$ is the $n$'th coefficient of some holomorphic cusp form, we prove a variety of Omega results for the twisted sum ${\sum }_{n\le x}{a}_n{e}^{2\pi in\alpha }$ and discuss their applications to the Ramanujan $\tau$-function and sums of $a_n$ over arithmetic progressions.