Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions

TL;DR

This paper improves the known distribution exponent of the divisor function $d_3(n)$ in prime arithmetic progressions from 2/3 to 8/11 using advanced bounds on Dirichlet L-functions.

Contribution

It advances the understanding of the distribution of $d_3(n)$ in prime moduli by applying the Petrow--Young subconvexity bound to achieve a higher exponent.

Findings

Distribution exponent improved from 2/3 to 8/11.

Utilizes Petrow--Young subconvexity bounds for Dirichlet L-functions.

Results hold uniformly over all residue classes modulo prime q.

Abstract

We say that $d_3(n)$ has exponent of distribution $\theta$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown, and Shparlinski, that after averaging over reduced residue classes $a \bmod q$, the function $d_3(n)$ has exponent of distribution $2/3$. Using the Petrow--Young subconvexity bound for Dirichlet $L$-functions, we improve this to $8/11$ when averaging over residue classes modulo a prime $q$.