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

TL;DR

The paper studies the distribution of the ternary divisor function $d_3(n)$ in arithmetic progressions and defines an exponent of distribution $\theta$ controlling uniform error terms for moduli $q \le x^{\theta-\varepsilon}$. Using the Petrow–Young subconvexity bound for Dirichlet $L$-functions within a smoothing/Voronoi-analytic framework, it improves the averaged exponent from $\theta=2/3$ to $\theta=8/11$ for prime moduli. The approach combines a Voronoi-type decomposition, Dirichlet $L$-function moment bounds, and Ramanujan-sum cancellation, with careful parameter optimization to achieve the bound $\sum_{a=1}^q |\Delta(a/q)|^2 \ll x^{3/2+\varepsilon} q^{11/16}$. This advances understanding of divisor-function distribution in progressions and demonstrates the efficacy of modern subconvexity results in averaging problems. The results may influence error terms in arithmetic-progressions and related multiplicative-number-theory estimates.

Abstract

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

Theorems & Definitions (22)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Theorem 1.2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof