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

Metin Can Aydemir; Muhammet Boran

arxiv: 2601.12601 · v2 · submitted 2026-01-18 · 🧮 math.NT

Improved Averaged Distribution of d₃(n) in Prime Arithmetic Progressions

Metin Can Aydemir , Muhammet Boran This is my paper

Pith reviewed 2026-05-16 13:31 UTC · model grok-4.3

classification 🧮 math.NT

keywords d_3(n)arithmetic progressionsdistribution exponentsubconvexityDirichlet L-functionsprime moduli

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The pith

Using subconvexity bounds, the averaged distribution exponent of d_3(n) in prime arithmetic progressions improves to 8/11.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an improved exponent of distribution for the three-divisor function d_3(n) when its values are averaged over residue classes modulo a prime. Previous work achieved an exponent of 2/3, but here the authors reach 8/11 by invoking a strong subconvexity estimate for Dirichlet L-functions. This larger exponent means the main term in the asymptotic formula for the sum of d_3(n) in such progressions remains valid for moduli as large as x to the power of nearly 8/11. A reader would care because stronger distribution results like this feed into applications on the distribution of primes and other arithmetic functions in short intervals and arithmetic progressions.

Core claim

We prove that d_3(n) has exponent of distribution 8/11 when averaging over reduced residue classes modulo a prime q, by applying the Petrow--Young subconvexity bound for Dirichlet L-functions to improve upon the prior exponent of 2/3 obtained by Nguyen.

What carries the argument

Petrow--Young subconvexity bound for Dirichlet L-functions, used to control the error terms in the averaging over prime moduli q.

If this is right

The asymptotic formula holds uniformly for all primes q up to x to the power 8/11 minus epsilon.
This extends the valid range of moduli beyond the earlier 2/3 exponent for averaged d_3(n).
The improvement applies specifically when the modulus is restricted to primes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Stronger future subconvexity bounds could push the distribution exponent above 8/11.
The restriction to prime moduli leaves open whether the same gain holds for composite q.
Similar averaging techniques might improve distribution exponents for other multiplicative functions.

Load-bearing premise

The Petrow-Young subconvexity bound for Dirichlet L-functions applies with sufficient uniformity in the ranges needed for the averaging argument over prime moduli q.

What would settle it

Finding a prime q near x to the power 8/11 where the averaged sum of d_3(n) over residue classes deviates from the main term by more than the allowed error would disprove the improved exponent.

read the original 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 $q$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Improves d3(n) distribution exponent to 8/11 over primes via Petrow-Young, but uniformity needs confirmation.

read the letter

The paper improves the averaged distribution exponent for d3(n) over prime moduli q from 2/3 to 8/11 by incorporating the Petrow-Young subconvexity bound. This is the key new result. The authors follow the averaging approach of Nguyen, Banks, Heath-Brown, and Shparlinski, but replace the previous bound with the stronger one from Petrow and Young. This gives a concrete numerical gain in the restricted setting where the moduli are primes. The argument is standard analytic number theory and the abstract states the improvement clearly. What could be soft is the uniformity question raised in the stress test. For the o(x/log x) error after summing over q ≤ x^{8/11-ε}, the subconvexity must apply uniformly when the conductor is about q(1+|t|) with |t| up to x^ε. The paper should include an explicit check that the cited theorem covers this range for prime q. Without that, the claimed exponent might need adjustment. This work is aimed at researchers in analytic number theory who study the distribution of multiplicative functions like d3(n) in arithmetic progressions. It would be of interest to those tracking improvements from subconvexity results. I would bring this to a reading group focused on number theory. I would not cite it in my own work unless I needed this specific exponent. It should go to peer review so the details on uniformity can be checked by experts.

Referee Report

1 major / 2 minor

Summary. The paper proves that the ternary divisor function d_3(n) has exponent of distribution 8/11 when averaged over reduced residue classes a mod q for prime q, improving on the previous exponent 2/3 obtained by Nguyen. The improvement is achieved by inserting the Petrow-Young subconvexity bound for Dirichlet L-functions into the character-sum estimates that arise after averaging.

Significance. If the uniformity of the Petrow-Young bound can be verified in the required hybrid range, the result advances the known distribution exponents for d_3(n) in prime arithmetic progressions. This has potential applications to sieve methods and the study of arithmetic functions in short intervals or APs with prime moduli. The manuscript correctly identifies the external subconvexity input as the source of the gain.

major comments (1)

[main averaging argument after Theorem 1.1] The main averaging argument (after the statement of the main theorem): the passage from the 2/3 exponent to 8/11 requires that the Petrow-Young bound supplies a uniform saving for prime moduli q up to x^{8/11-ε} in the hybrid range where the analytic conductor is q(1+|t|) with |t| ≪ x^ε. The manuscript does not record an explicit verification that the cited theorem applies throughout this rectangle without loss of the necessary saving; this uniformity is load-bearing for the claimed improvement.

minor comments (2)

[Introduction] The introduction should include a brief comparison table or explicit statement of the previous exponent 2/3 from Nguyen together with the new 8/11 result.
[Section 2] Notation for the averaged sum over a mod q should be defined once at the beginning of the proof section rather than re-introduced.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to explicitly confirm the uniformity of the Petrow-Young bound in the hybrid range required for the 8/11 exponent. We address this point below and will incorporate the verification into the revised manuscript.

read point-by-point responses

Referee: [main averaging argument after Theorem 1.1] The main averaging argument (after the statement of the main theorem): the passage from the 2/3 exponent to 8/11 requires that the Petrow-Young bound supplies a uniform saving for prime moduli q up to x^{8/11-ε} in the hybrid range where the analytic conductor is q(1+|t|) with |t| ≪ x^ε. The manuscript does not record an explicit verification that the cited theorem applies throughout this rectangle without loss of the necessary saving; this uniformity is load-bearing for the claimed improvement.

Authors: We agree that an explicit verification is desirable. The Petrow-Young subconvexity bound (as stated in the cited reference) holds uniformly for the hybrid conductor q(1+|t|) throughout the rectangle q ≤ X, |t| ≤ X^ε for any fixed ε>0 and X sufficiently large. In our setting, after the main averaging step, we have q ≤ x^{8/11-ε} (prime) and |t| ≪ x^ε, so the analytic conductor is ≪ x^{8/11-ε/2}. The saving supplied by the bound is therefore uniform and of the required strength to upgrade the exponent from 2/3 to 8/11. We will add a short paragraph immediately after the statement of Theorem 1.1 that records this range check and confirms that no loss of saving occurs. revision: yes

Circularity Check

0 steps flagged

No circularity: improvement obtained by applying external Petrow-Young subconvexity bound

full rationale

The manuscript derives the improved exponent 8/11 for the averaged distribution of d_3(n) in prime arithmetic progressions by substituting the Petrow-Young subconvexity bound into the character-sum estimates that appear after averaging over residue classes a mod q with q prime. The bound itself is cited from independent prior work and is not obtained by fitting parameters inside the present paper, nor by any self-citation chain that reduces the central claim to a tautology. The derivation therefore remains self-contained once the external bound is granted; no equation or step collapses by construction to an input defined within the manuscript. The skeptic's uniformity concern pertains to correctness of application rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Dirichlet L-functions and the external Petrow-Young subconvexity bound; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Petrow-Young subconvexity bound holds for Dirichlet L-functions in the ranges required for the averaging
Invoked directly to obtain the improved 8/11 exponent in the prime-moduli averaging argument.

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