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

Abstract

We obtain asymptotics with a power saving error term for ${\int }_0^1|{\sum }_{n\le X}d(n)e(n\alpha ){|}^{s}d\alpha$ for $s<2$, where $d(n)={\sum }_{d|n}1$ is the divisor function. We also obtain such asymptotics for ${\int }_0^1|{\sum }_{n\le X}{\lambda }_f(n)e(n\alpha ){|}^{s}d\alpha$ for all $s>0$. Our proof an iterative method with repeated applications of Jutila's variant of the circle method and Voronoi summation.

20 pages