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

1. Problem Statement

The paper addresses the distribution of the ternary divisor function $d_3(n)$ (which counts the number of ways to write $n$ as a product of three positive integers) in arithmetic progressions.

Specifically, the authors investigate the exponent of distribution $\theta$. A function $f(n)$ is said to have an exponent of distribution $\theta$ if the error term in the summation of $f(n)$ over arithmetic progressions $n \equiv a \pmod q$ is small uniformly for moduli $q$ up to $x^{\theta - \epsilon}$.

• Historical Context:
  • For the standard divisor function $d(n)$, the exponent is known to be $2/3$ (Selberg, Linnik, Hooley).
  • For $d_3(n)$, the classical bound was $1/2$.
  • Averaged Results: Banks, Heath-Brown, and Shparlinski introduced averaging over residue classes to improve these bounds. Nguyen (2005) proved that for $d_3(n)$, the averaged exponent of distribution is $2/3$.
  • Recent Development: Parry (2024) improved the averaged exponent for the quaternary divisor function $d_4(n)$ to $4/7$ using subconvexity bounds.

The Goal: The authors aim to improve the averaged exponent of distribution for $d_3(n)$ from the previous best of $2/3$ to $8/11$ (where $8/11 \approx 0.727$), specifically when averaging over residue classes modulo a prime $q$.

2. Methodology

The proof relies on a combination of analytic number theory techniques, specifically the Voronoi summation formula and subconvexity bounds for Dirichlet $L$-functions.

A. Setup and Definitions

The authors define the error term $\Delta(a/q)$ as the difference between the sum of $d_3(n)e(na/q)$ and its main term (residue at $s=1$). The goal is to bound the second moment of this error term averaged over $a$:
$$\sum_{a=1}^q |\Delta(a/q)|^2$$

B. Voronoi Summation and Spectral Decomposition

Using a smoothed version of the sum (Lemma 1, based on Ivić's work), the error term $\tilde{\Delta}(h/q)$ is transformed into a dual sum involving:

1. Bessel-like functions: $U(x)$ and $V(x)$ derived from Gamma functions.
2. Exponential Sums: Coefficients $A_{h/q}(n)$ and $B_{h/q}(n)$, which are sums of additive characters involving the divisor structure.

The core of the method involves analyzing the sum of squares of these error terms, which reduces to estimating sums of the form:
$$\sum_{n, m} d_3(n)d_3(m) c_q(n-m) \hat{w}_q(n)\hat{w}_q(m)$$
where $c_q(\cdot)$ is the Ramanujan sum and $\hat{w}_q$ is the transform of the smoothing weight.

C. Key Analytic Tools

1. Petrow–Young Subconvexity Bound:
   The authors utilize the breakthrough subconvexity bound for Dirichlet $L$-functions by Petrow and Young (2023):
   $$L(1/2+it, \chi) \ll_\epsilon (q(1+|t|))^{1/6+\epsilon}$$
   This bound is crucial for controlling the third moment of $L$-functions, which appears when applying Perron's formula to the divisor sums.

2. Moment Estimates:

   • Lemma 6 & 7: The authors establish bounds for the second and third moments of $L$-functions on the critical line. Specifically, Lemma 7 proves:
     $$\sum_{\chi \neq \chi_0} \left( \int_1^T |L(1/2+it, \chi)|^3 \frac{dt}{t} \right)^2 \ll_\epsilon q^{11/8}(qT)^\epsilon$$
     This $q^{11/8}$ factor is the critical component that allows the exponent to exceed $2/3$.

3. Combinatorial Bounds on Exponential Sums:

   • Lemma 2 & 3: The authors derive precise bounds for the correlation of the coefficients $A_{h/q}(n)$. They show that for prime $q$, the sum over $h$ of $A_{h/q}(n)A_{h/q}(m)$ relates to Ramanujan sums $c_q(n-m)$ and $c_q(n+m)$.
   • This allows the transformation of the problem into bounding sums involving $d_3(n)d_3(m)c_q(n-m)$.

4. Lemma 10 & 11 (The Core Estimate):
   By combining the $L$-function moment bounds with the Voronoi dual sums, the authors prove a bound for the correlation sum:
   $$\sum_{n,m} d_3(n)d_3(m)c_q(n-m) \ll \frac{NM}{q} + q^{11/8}\sqrt{NM} + \dots$$
   This bound is then applied to the smoothed sums in the proof of the main theorem.

3. Key Contributions

1. Improvement of the Exponent: The paper successfully raises the averaged exponent of distribution for $d_3(n)$ from $2/3$ to $8/11$.
2. Application of Petrow–Young Bound: This is one of the first applications of the Petrow–Young subconvexity bound to the distribution of divisor functions in arithmetic progressions, demonstrating its power in extending the range of moduli.
3. Refined Analysis of $d_3(n)$: The authors provide a detailed analysis of the third moment of $L$-functions in the context of the ternary divisor function, bridging the gap between the methods used for $d(n)$ and $d_4(n)$.

4. Main Results

Theorem 1.1:
Let $q$ be a prime and $x$ be large. For any $\epsilon > 0$, the sum of the squared error terms satisfies:
$$\sum_{a=1}^q |\Delta(a/q)|^2 \ll_\epsilon x^{3/2+\epsilon} q^{11/16}$$
This inequality holds uniformly for all primes $q \le x^{8/11 - \epsilon}$.

Theorem 1.2 (Corollary):
By applying the Cauchy–Schwarz inequality to Theorem 1.1, the authors derive the bound for the first moment:
$$\sum_{a=1}^q |E_x(q, a)| \ll_\epsilon x^{3/4+\epsilon} q^{11/32}$$
This confirms that $d_3(n)$ is equidistributed on average for moduli up to $x^{8/11}$.

5. Significance

• Theoretical Advancement: The result pushes the boundary of what is known about the distribution of divisor functions. The exponent $8/11$ is significantly larger than the classical $2/3$, indicating a much stronger cancellation of error terms when averaging over residue classes.
• Methodological Impact: The paper validates the utility of the Petrow–Young subconvexity bound in problems involving higher-order divisor functions. It suggests that similar techniques could be applied to $d_k(n)$ for $k > 3$ to further improve their distribution exponents.
• Prime Moduli Restriction: The result is currently specific to prime moduli. This restriction is necessary to utilize the specific properties of Ramanujan sums and the Petrow–Young bound (which applies to cube-free moduli, including primes) in the way the authors structured the proof. Extending this to composite moduli remains an open challenge.

In summary, Aydemir and Boran have successfully leveraged modern subconvexity results to refine the understanding of how the ternary divisor function distributes itself across arithmetic progressions, achieving a new record for the averaged level of distribution.