A note on the distribution of $d_3(n)$ in arithmetic progressions

TL;DR

This paper investigates the distribution of the 3-fold divisor function in arithmetic progressions, improving bounds on its average distribution exponent compared to previous results.

Contribution

It provides stronger bounds on the average distribution of the 3-fold divisor function in arithmetic progressions, building on and extending prior work.

Findings

Improved bounds on the distribution exponent of $d_3(n)$

Enhanced understanding of divisor function behavior in progressions

Extension of previous distribution results

Abstract

Nguyen has shown that on averaging over $a=1,...,q$ the 3-fold divisor function has exponent of distribution 2/3, following \cite {banks}. We follow [2] which leads to stronger bounds.