On Signs of Fourier Coefficients on GL( <i>n</i> )

Abstract We study statistical properties of Fourier coefficients of automorphic forms on $\textrm{GL}(n)$. 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 of Jääsaari obtained in the case of self-dual forms on $\textrm{GL}(3)$ to almost all forms on $\textrm{GL}(n),$ and our method moreover circumvents the assumption of the Generalized Ramanujan Conjecture.