On the distribution of additive twists of the divisor function and Hecke eigenvalues

Abstract

We obtain asymptotics with a power saving error term for $\int_0^1|\sum_{n\le X} d(n)e(n\alpha)|^sd\alpha$ for $s < 2$, where $d(n) = \sum_{d | n} 1$ is the divisor function. We also obtain such asymptotics for $\int_0^1|\sum_{n\le X}\lambda_f(n)e(n\alpha)|^sd\alpha$ for all $s > 0$. Our proof an iterative method with repeated applications of Jutila's variant of the circle method and Voronoi summation.