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

TL;DR

This paper derives precise asymptotic formulas with power-saving error terms for moments of additive twists of the divisor function and Hecke eigenvalues, advancing understanding of their distribution.

Contribution

It introduces an iterative method combining Jutila's circle method and Voronoi summation to analyze additive twists of divisor functions and Hecke eigenvalues.

Findings

Asymptotics with power-saving error terms for moments of divisor function twists.

Asymptotics for moments of Hecke eigenvalue twists for all s > 0.

Development of an iterative proof technique using circle method and Voronoi summation.

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.