Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

Abstract

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a suitable range, generalizing the case N=2 treated by E. Fouvry, S. Ganguly, E. Kowalski and P. Michel. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.