Exponential sums related to Maass forms
We estimate short exponential sums weighted by the Fourier coefficients of a Maass form and discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct, we 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, 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 use the approximate functional equation to bound somewhat longer short exponential sums.