On Short Sums Involving Fourier Coefficients of Maass Forms

TL;DR

This paper investigates short sums of Fourier coefficients of Maass forms for SL(n,Z) with n≥3, providing asymptotic evaluations of their second moments under certain hypotheses, advancing understanding of automorphic forms.

Contribution

It offers the first asymptotic evaluation of the second moment of short sums of Fourier coefficients of Maass forms for higher rank groups under generalized Lindel"of and Ramanujan-Petersson assumptions.

Findings

Asymptotic formula for the second moment of short sums

Conditional results based on generalized Lindel"of hypothesis

Extension of techniques to higher rank automorphic forms

Abstract

We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindel\"of hypothesis and a slightly stronger upper bound concerning the exponent towards the Ramanujan-Petersson conjecture that is currently known. In particular, in this case we evaluate the second moment of the sums in question asymptotically.