On signs of Fourier coefficients on GL(n)

TL;DR

This paper investigates the statistical behavior of Fourier coefficients of automorphic forms on GL(n), revealing their nonvanishing, sign change patterns, and asymptotic densities without relying on the Generalized Ramanujan Conjecture.

Contribution

It extends previous results on GL(3) to general GL(n), providing new asymptotic formulas and sign change results for Fourier coefficients of automorphic forms.

Findings

Asymptotic count of nonvanishing Fourier coefficients

Positive proportion of sign changes among real coefficients

Asymptotic density of sign patterns

Abstract

We study statistical properties of Fourier coefficients of automorphic forms on GL(n). For most Hecke-Maass cusp forms, we give the asymptotic number of nonvanishing coefficients, show that there is a positive proportion of sign changes among them, when these are real, and describe the asymptotic density of these signs. We generalize the results by J\"a\"asaari obtained in the case of self-dual forms of GL(3) and our method moreover circumvents the assumption of the Generalized Ramanujan Conjecture.