Limiting Distribution and Rate of Convergence for GL(3) Fourier Coefficients

Abstract

In a work of Heath-Brown, it is proved that in the Pilz divisor problem, the normalized error term $Δ_3(x)$ has a distribution function. In this paper, we prove an analogue of this result in the setting of GL(3). For a given self-dual GL(3) Hecke--Maass cusp form $f$ with normalized Fourier coefficients $A_f(n,m)$, let $Δ_f(x)=\sum_{n\leqslant x}A_f(n,1)$. We show that the function $x^{-1/3}Δ_f(x)$ has a distribution function and we obtain a quantitative rate of convergence for the limiting distribution.