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

TL;DR

This paper extends the understanding of distribution functions and convergence rates for Fourier coefficients of GL(3) automorphic forms, analogous to previous results in divisor problems.

Contribution

It proves the existence of a distribution function for normalized sums of GL(3) Fourier coefficients and provides a quantitative rate of convergence.

Findings

Normalized sums of GL(3) Fourier coefficients have a distribution function.

Established a quantitative rate of convergence for the limiting distribution.

Abstract

In a work of Heath-Brown, it is proved that in the Pilz divisor problem, the normalized error term $\Delta_3(x)$ 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 $\Delta_f(x)=\sum_{n\leqslant x}A_f(n,1)$. We show that the function $x^{-1/3}\Delta_f(x)$ has a distribution function and we obtain a quantitative rate of convergence for the limiting distribution.