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

In a work of Heath-Brown, it is proved that in the Pilz divisor problem, the normalized error term ${\mathrm{\Delta }}_3(x)$ 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 ${\mathrm{\Delta }}_f(x)={\sum }_{n⩽x}{A}_f(n,1)$. We show that the function $x^{-1\mathrm{/}3}{\mathrm{\Delta }}_f(x)$ has a distribution function and we obtain a quantitative rate of convergence for the limiting distribution.