Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

TL;DR

This paper demonstrates that the divisor function and Hecke eigenvalues in arithmetic progressions follow Gaussian distributions in certain ranges, revealing new probabilistic behaviors of these classical arithmetic functions.

Contribution

It establishes Gaussian distribution results for the divisor function and Hecke eigenvalues in residue classes, using moment asymptotics and monodromy group independence.

Findings

Divisor function in residue classes follows a Gaussian distribution in a restricted range.

Hecke eigenvalues exhibit similar Gaussian distribution behavior.

Joint distribution of these functions in related residue classes is characterized.

Abstract

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.