Poisson-Dirichlet approximation for counting integers with divisors in an interval

Tony Haddad

arxiv: 2512.13669 · v2 · submitted 2025-12-15 · 🧮 math.NT · math.PR

Poisson-Dirichlet approximation for counting integers with divisors in an interval

Tony Haddad This is my paper

Pith reviewed 2026-05-16 21:58 UTC · model grok-4.3

classification 🧮 math.NT math.PR

keywords asymptotic formulasdivisor countingprime factorizationArratia couplingconvex setsboundary probabilitiesPoisson-Dirichletnumber theory

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The pith

A convex-set comparison inequality combined with Arratia coupling reduces counting integers with restricted prime factorizations to bounding two boundary proximity probabilities.

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

The paper develops a general method to obtain asymptotic formulas for the number of integers whose prime factorizations lie inside any chosen subset of ell-one space. It proves a simple inequality that compares the laws of two random variables valued inside a convex subset of a normed vector space, then pairs this inequality with a refined version of Arratia's coupling. The combination reduces the original counting problem to the task of bounding two probabilities that measure closeness to the boundary of the subset. The method is carried through for the concrete case of integers up to x that possess a divisor inside an interval (y, z) whenever the ratio z over y tends to infinity, producing an explicit asymptotic formula. A sympathetic reader cares because many divisor and factorization counting questions in number theory become tractable once they are translated into these boundary estimates.

Core claim

By establishing a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space and combining it with the refined Arratia coupling, the work shows that an asymptotic formula for the number of integers whose prime factorization lies in any given subset of ell-one real space can be reduced to bounding two key probabilities that measure proximity to the boundary of the subset. When applied to the count of integers in [1, x] having a divisor in (y, z) in the regime where z/y tends to infinity, this reduction supplies the desired asymptotic formula.

What carries the argument

A simple inequality comparing the laws of two random variables valued in a convex subset of a normed vector space, used together with the refined Arratia coupling to convert counting problems into boundary-probability estimates.

If this is right

An explicit asymptotic formula holds for the number of integers up to x possessing a divisor in (y, z) whenever z/y tends to infinity.
The same reduction strategy applies to any other subset of ell-one real space once the two boundary proximity probabilities can be bounded.
Asymptotics become available for many further counting problems defined by conditions on prime factorizations, provided the boundary probabilities remain controllable.

Where Pith is reading between the lines

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

The method may extend to regimes in which z/y tends to a positive constant or to zero, provided adjusted boundary estimates are supplied.
Similar boundary-probability reductions could apply to related problems such as counting integers with all prime factors in prescribed ranges.
The explicit asymptotic could be used to test the accuracy of Poisson-Dirichlet approximations in finite-x regimes through direct comparison.

Load-bearing premise

The refined Arratia coupling applies directly to the convex-set comparison inequality without introducing error terms that would disturb the leading asymptotic term when the interval ratio tends to infinity.

What would settle it

Direct numerical computation of the count of integers up to a large x with a divisor in (y, z) for several sequences where z/y grows, compared against the predicted main term; a persistent mismatch of size larger than the error term would disprove the claim.

read the original abstract

We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia's coupling, recently refined by Koukoulopoulos and the author, we obtain a general strategy to reduce the problem of finding an asymptotic formula for the number of integers whose prime factorization lies in any given subset of $\ell^1(\mathbb R)$, to bounding two key probabilities measuring proximity to the boundary of the subset in question. We apply this strategy to obtain an asymptotic formula for counting integers in $[1, x]$ that have a divisor in an interval $(y, z)$ in the regime $z/y \to \infty$ as $x \to \infty$.

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

The new convex-set inequality plus refined coupling reduces the divisor-interval count to boundary probabilities, but the error transfer in the z/y to infinity regime is the part that needs checking.

read the letter

The core contribution is a straightforward inequality comparing the laws of two random variables on a convex subset of a normed space. Paired with the author's earlier refinement of Arratia's coupling, it turns the problem of counting integers whose normalized prime exponents land in a given subset of ℓ¹ into the task of bounding two boundary probabilities. They use this to derive an asymptotic for the number of n ≤ x with a divisor in (y, z) when z/y → ∞. That reduction strategy is the genuinely new piece and looks reusable for other prime-factor subset problems. The inequality itself is stated independently, so there is no immediate circularity with the prior coupling work. The application to the widening-interval divisor count is also new on its face. The soft spot is whether the coupling error stays small enough once the target set is the preimage of (y, z) and the ratio diverges. The prior coupling was built for a different class of sets; if the total-variation or Wasserstein discrepancy only improves like 1/log log x or slower in this regime, it could leak into the main term and spoil the claimed asymptotic. The abstract asserts that the combination works cleanly, but the letter needs the explicit bounds to confirm the boundary terms really absorb everything. This is for analytic number theorists who already use probabilistic models for divisor and factorization counts. Someone working on similar subset problems would find the reduction useful even if they have to adapt the error estimates. It is worth sending to a serious referee who can verify the coupling transfer and the concrete bounds in the z/y → ∞ case.

Referee Report

2 major / 2 minor

Summary. The paper introduces a comparison inequality for the distributions of two random variables taking values in a convex subset of a normed vector space. By combining this inequality with a refined Arratia coupling (from prior work with Koukoulopoulos), it reduces the problem of obtaining an asymptotic for the number of integers whose normalized prime-exponent vectors lie in a given subset of ℓ¹(ℝ) to bounding two boundary-proximity probabilities. The method is applied to derive an asymptotic formula for the count of integers ≤ x possessing a divisor in the interval (y, z) in the regime z/y → ∞.

Significance. If the coupling error is controlled so that it is absorbed into the boundary terms, the reduction supplies a general and flexible strategy for a range of counting problems involving subsets of prime factorizations. The explicit reduction to boundary probabilities is a clear organizational strength and could extend to other divisor-distribution questions in analytic number theory.

major comments (2)

[Introduction and §3 (general strategy)] The central reduction (stated after the convex-set inequality and before the application) asserts that the refined Arratia coupling produces no additional error beyond the two boundary probabilities. Because the coupling was originally established for a different class of sets, an explicit bound is required showing that the total-variation or Wasserstein distance remains o(1) uniformly for the preimage sets corresponding to divisors in (y, z) when z/y → ∞; without this, the main asymptotic term may be polluted.
[§4 (application)] In the application to the divisor-counting problem (the theorem stated after the boundary-probability estimates), the claimed asymptotic is obtained once the two boundary probabilities are shown to be o(1). The manuscript must supply the explicit verification that these probabilities are indeed o(1) in the z/y → ∞ regime, together with the precise error term that results from the combination of the inequality and the coupling.

minor comments (2)

[§4] Notation for the normalized exponent vector and the precise definition of the convex set should be recalled at the beginning of the application section for readability.
[Introduction] A short remark comparing the obtained error term with existing results on divisor counts in short intervals would help situate the new asymptotic.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful comments and positive evaluation of the paper's general strategy. We address each major comment below and will make the necessary revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Introduction and §3 (general strategy)] The central reduction (stated after the convex-set inequality and before the application) asserts that the refined Arratia coupling produces no additional error beyond the two boundary probabilities. Because the coupling was originally established for a different class of sets, an explicit bound is required showing that the total-variation or Wasserstein distance remains o(1) uniformly for the preimage sets corresponding to divisors in (y, z) when z/y → ∞; without this, the main asymptotic term may be polluted.

Authors: We agree that an explicit bound on the coupling error is needed for the specific sets arising in the divisor problem. In the revised manuscript, we will include a detailed verification in §3 showing that the total variation distance between the coupled variables is o(1) uniformly in the regime z/y → ∞ for these preimage sets. This will be based on the refined Arratia coupling from our prior work and the specific geometry of the sets defined by having a divisor in (y,z). revision: yes
Referee: [§4 (application)] In the application to the divisor-counting problem (the theorem stated after the boundary-probability estimates), the claimed asymptotic is obtained once the two boundary probabilities are shown to be o(1). The manuscript must supply the explicit verification that these probabilities are indeed o(1) in the z/y → ∞ regime, together with the precise error term that results from the combination of the inequality and the coupling.

Authors: We will add the explicit verification that the two boundary probabilities are o(1) as z/y → ∞, including the necessary estimates in §4. Furthermore, we will state the main theorem with the precise error term obtained from combining the comparison inequality and the coupling, ensuring the asymptotic formula is fully justified with the error terms made explicit. revision: yes

Circularity Check

0 steps flagged

New convex-set inequality and cited prior coupling yield non-circular reduction to boundary probabilities

full rationale

The paper introduces an original simple inequality comparing laws of random variables in convex subsets of a normed space. This is combined with the refined Arratia coupling established in prior work by Koukoulopoulos and the author. The resulting strategy genuinely reduces the asymptotic count to bounding two boundary-proximity probabilities, without any equation or claim that equates the final asymptotic to its inputs by construction. The self-citation supports an external tool (the coupling) rather than serving as the sole justification for the central claim. No self-definitional loop, fitted-input prediction, or ansatz smuggling occurs. The derivation remains self-contained with independent content from the new inequality.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the existence and properties of the Poisson-Dirichlet process for prime-factor distributions and on the validity of the refined Arratia coupling from prior work; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The refined Arratia coupling controls the joint distribution of prime factors sufficiently well for convex-set comparisons.
Invoked to combine the new inequality with existing coupling results.

pith-pipeline@v0.9.0 · 5413 in / 1302 out tokens · 34727 ms · 2026-05-16T21:58:08.662944+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia’s coupling... we obtain a general strategy to reduce... to bounding two key probabilities measuring proximity to the boundary
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3... νx(A)−μ(A) ≪ P⋆[{Primes(Nx,x),V}⊆∂max{Sx,Tx}A] + 1/log x

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

On the amount of dependence in the prime factorization of a random integer

R. Arratia – “On the amount of dependence in the prime factorization of a random integer”, inContemporary Combinatorics, Bolyai Soc. Math. Stud., vol. 10, J´anos Bolyai Math. Soc., Budapest, 2002, p. 29–91

work page 2002
[2]

On the density of certain sequences of integers

A. S. Besicovitch – “On the density of certain sequences of integers”,Math. Ann.110(1934), p. 336–341

work page 1934
[3]

On the distribution of large prime divisors

P. Billingsley – “On the distribution of large prime divisors”,Period. Math. Hungar.2(1972), p. 283–289

work page 1972
[4]

Note on the sequences of integers no one of which is divisible by any other

P. Erd ˝os – “Note on the sequences of integers no one of which is divisible by any other”,J. London Math. Soc. 10(1935), p. 126–128

work page 1935
[5]

A generalization of a theorem of Besicovitch

P. Erd ˝os – “A generalization of a theorem of Besicovitch”,J. London Math. Soc.11(1936), p. 92–98

work page 1936
[6]

Feng -The Poisson–Dirichlet distribution and related topics

S. Feng -The Poisson–Dirichlet distribution and related topics. Models and asymptotic behaviors, Probability and its Appl., Springer, Heidelberg, 2010

work page 2010
[7]

The distribution of integers with a divisor in a given interval

K. Ford – “The distribution of integers with a divisor in a given interval”,Ann. of Math. (2)168(2008), no. 2, p. 367–433

work page 2008
[8]

On Arratia’s coupling and the Dirichlet law for the factors of a random integer

T. Haddad & D. Koukoulopoulos – “On Arratia’s coupling and the Dirichlet law for the factors of a random integer”,J. ´Ec. polytech., Math.12(2025), p. 1565–1604

work page 2025
[9]

J. F. C. Kingman –Poisson processes, Oxford Studies of Probab., vol. 3, The Clarendon Press, Oxford University Press, New York, 1993

work page 1993
[10]

Analysis of a simple factorization algorithm

D. E. Knuth & L. Trabb Pardo – “Analysis of a simple factorization algorithm”,Theoret. Comput. Sci.3(1976), no. 3, p. 321–348

work page 1976
[11]

Lois de r ´epartition des diviseurs, 2

G. Tenenbaum – “Lois de r ´epartition des diviseurs, 2”,Acta Arithm.38(1980), p. 1–36

work page 1980
[12]

Sur la probabilit ´e qu’un entier poss `ede un diviseur dans un intervalle donn ´e

G. Tenenbaum – “Sur la probabilit ´e qu’un entier poss `ede un diviseur dans un intervalle donn ´e”,Compositio Math.51(1984), no. 2, p. 243–263

work page 1984
[13]

A rate estimate in Billingsley’s Theorem for the size distribution of large prime factors

G. Tenenbaum – “A rate estimate in Billingsley’s Theorem for the size distribution of large prime factors”,Q. J. Math.51(2000), no. 3, p. 385–403. DEPARTEMENT OFMATHEMATICS ANDSTATISTICS, UNIVERSITY OFTURKU, 20014 TURKU, FINLAND Email address:tohadd@utu.fi

work page 2000

[1] [1]

On the amount of dependence in the prime factorization of a random integer

R. Arratia – “On the amount of dependence in the prime factorization of a random integer”, inContemporary Combinatorics, Bolyai Soc. Math. Stud., vol. 10, J´anos Bolyai Math. Soc., Budapest, 2002, p. 29–91

work page 2002

[2] [2]

On the density of certain sequences of integers

A. S. Besicovitch – “On the density of certain sequences of integers”,Math. Ann.110(1934), p. 336–341

work page 1934

[3] [3]

On the distribution of large prime divisors

P. Billingsley – “On the distribution of large prime divisors”,Period. Math. Hungar.2(1972), p. 283–289

work page 1972

[4] [4]

Note on the sequences of integers no one of which is divisible by any other

P. Erd ˝os – “Note on the sequences of integers no one of which is divisible by any other”,J. London Math. Soc. 10(1935), p. 126–128

work page 1935

[5] [5]

A generalization of a theorem of Besicovitch

P. Erd ˝os – “A generalization of a theorem of Besicovitch”,J. London Math. Soc.11(1936), p. 92–98

work page 1936

[6] [6]

Feng -The Poisson–Dirichlet distribution and related topics

S. Feng -The Poisson–Dirichlet distribution and related topics. Models and asymptotic behaviors, Probability and its Appl., Springer, Heidelberg, 2010

work page 2010

[7] [7]

The distribution of integers with a divisor in a given interval

K. Ford – “The distribution of integers with a divisor in a given interval”,Ann. of Math. (2)168(2008), no. 2, p. 367–433

work page 2008

[8] [8]

On Arratia’s coupling and the Dirichlet law for the factors of a random integer

T. Haddad & D. Koukoulopoulos – “On Arratia’s coupling and the Dirichlet law for the factors of a random integer”,J. ´Ec. polytech., Math.12(2025), p. 1565–1604

work page 2025

[9] [9]

J. F. C. Kingman –Poisson processes, Oxford Studies of Probab., vol. 3, The Clarendon Press, Oxford University Press, New York, 1993

work page 1993

[10] [10]

Analysis of a simple factorization algorithm

D. E. Knuth & L. Trabb Pardo – “Analysis of a simple factorization algorithm”,Theoret. Comput. Sci.3(1976), no. 3, p. 321–348

work page 1976

[11] [11]

Lois de r ´epartition des diviseurs, 2

G. Tenenbaum – “Lois de r ´epartition des diviseurs, 2”,Acta Arithm.38(1980), p. 1–36

work page 1980

[12] [12]

Sur la probabilit ´e qu’un entier poss `ede un diviseur dans un intervalle donn ´e

G. Tenenbaum – “Sur la probabilit ´e qu’un entier poss `ede un diviseur dans un intervalle donn ´e”,Compositio Math.51(1984), no. 2, p. 243–263

work page 1984

[13] [13]

A rate estimate in Billingsley’s Theorem for the size distribution of large prime factors

G. Tenenbaum – “A rate estimate in Billingsley’s Theorem for the size distribution of large prime factors”,Q. J. Math.51(2000), no. 3, p. 385–403. DEPARTEMENT OFMATHEMATICS ANDSTATISTICS, UNIVERSITY OFTURKU, 20014 TURKU, FINLAND Email address:tohadd@utu.fi

work page 2000