Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms

TL;DR

This paper provides an estimate for the mean square of short exponential sums involving Fourier coefficients of cusp forms, with applications to understanding their average behavior over short intervals.

Contribution

It introduces a new method for estimating the mean square of short exponential sums with cusp form coefficients, focusing on relatively short averaging intervals.

Findings

Derived a bound for the mean square of the sums

Showed the effectiveness of the method for short intervals

Enhanced understanding of Fourier coefficient behavior in short sums

Abstract

We estimate the mean square of a short exponential sum involving Fourier coefficients of a cusp form with a linear twist, a smooth weight function, and a relatively short averaging interval.