Exponential Sums Related to Maass Forms

TL;DR

This paper develops new estimates for short exponential sums involving Maass form coefficients, improving bounds for long sums and extending techniques to non-linear sums, with implications for analytic number theory.

Contribution

It introduces a transformation formula for non-linear exponential sums and extends short sum estimates to Maass and holomorphic cusp forms, enhancing existing bounds.

Findings

Improved bounds for long linear sums of Maass form coefficients

Extended short sum estimates for holomorphic cusp forms

Derived an analogue of the approximate functional equation

Abstract

We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients. As an application of these, we remove the logarithm from the classical upper bound for long linear sums weighted by Fourier coefficients of Maass forms, the resulting estimate being the best possible. This also involves improving the upper bounds for long linear sums with rational additive twists, the gains again allowed by the estimates for the short sums. Finally, we shall use the approximate functional equation to bound somewhat longer short exponential sums.