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 $${N \geq 2}$$ , satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.