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arxiv: 2605.01155 · v1 · submitted 2026-05-01 · 🧮 math.NT

Sets of integers satisfying Bateman-Horn statistics

William Banks , Kevin Ford This is my paper

Pith reviewed 2026-05-09 17:58 UTC · model grok-4.3

classification 🧮 math.NT
keywords Bateman-Horn conjecturerandom setssingular seriesadmissible tuplesasymptoticsprobabilistic number theoryinteger distributions
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The pith

Certain random sets of integers almost surely satisfy the full Bateman-Horn asymptotics when membership in the set replaces primality.

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

The paper establishes that random subsets of the positive integers, generated according to suitable probability models, obey with probability one the precise counting asymptotics conjectured by Bateman and Horn for admissible tuples of polynomials. The condition that all polynomial values are prime is replaced by the condition that all values lie inside the random set, and the asymptotics hold with a strong error term. A reader would care because this shows that Bateman-Horn-type statistics are generic rather than special to the primes, implying that many different subsets of the integers exhibit the same distribution properties that the primes are conjectured to have.

Core claim

We prove that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where we have replaced 'f_1(n), ..., f_k(n) are all prime' with 'f_1(n), ..., f_k(n) all lie in the random set.' In particular, sets of integers satisfying Bateman-Horn are plentiful.

What carries the argument

Random sets of integers whose elements are included independently with densities chosen so that expected tuple counts match the singular series; this mechanism allows probabilistic arguments to establish the almost-sure asymptotics.

If this is right

  • Many subsets of the integers, not just the primes, exhibit the same counting behavior for polynomial tuples.
  • The Bateman-Horn asymptotics can be realized by random models without any reference to primality.
  • Such random sets exist in abundance, so properties derived from Bateman-Horn statistics apply to a large class of sets.
  • The proofs establish the result simultaneously for all admissible tuples and all x, with a strong uniform error term.

Where Pith is reading between the lines

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

  • The same random-set technique might be applied to other conjectures that predict prime-like counts, such as Hardy-Littlewood or Schinzel hypotheses.
  • Numerical experiments could sample finite random sets according to the model and directly verify the asymptotic counts for small tuples.
  • If the primes themselves can be approximated by limits of such random sets, the result would give heuristic support for the original Bateman-Horn conjecture.
  • The approach separates the arithmetic structure of the polynomials from the randomness of the ambient set.

Load-bearing premise

The probability model generating the random set must be tuned so that the expected number of n where all f_i(n) belong to the set exactly matches the singular-series prediction of the Bateman-Horn conjecture.

What would settle it

An explicit probability model satisfying the density condition for which there exists an admissible polynomial tuple such that, with positive probability, the counting function deviates from the predicted main term by more than the claimed error term for infinitely many x.

read the original abstract

In 1962, Bateman and Horn conjectured precise asymptotics for the count of positive integers n \le x for which f_1(n), ..., f_k(n) are all prime, where (f_1, ..., f_k) is an admissible k-tuple of polynomials in one variable. We prove that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where we have replaced "f_1(n), ..., f_k(n) are all prime" with "f_1(n), ..., f_k(n) all lie in the random set." In particular, sets of integers satisfying Bateman-Horn are plentiful.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where the condition that f1(n),...,fk(n) are all prime is replaced by the condition that f1(n),...,fk(n) all lie in the random set, for any admissible k-tuple of polynomials. In particular, such sets are shown to be plentiful.

Significance. If the central claim holds, the result demonstrates that Bateman-Horn statistics arise generically for appropriately constructed random sets rather than being special to the primes, thereby providing a rigorous probabilistic model supporting the plausibility of the classical conjecture. The almost-sure statement together with a strong error term is a notable strength.

major comments (1)
  1. [probability model construction] The probability model for the random set (presumably defined in the section introducing the construction) must be shown to produce an expected count that includes the full singular series factor ∏_p (1−1/p)^k (1−ω(p)/p). An independent Bernoulli model with P(m ∈ S) ∼ c/log m would yield only the product of marginal probabilities, omitting the Euler product and reducing to the S=1 case, which contradicts the claim of matching the full Bateman-Horn form.
minor comments (1)
  1. The abstract is concise but the introduction would benefit from an explicit statement of the precise density and dependence structure used for the random set, to allow immediate comparison with classical random models such as Cramér's.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The major comment raises an important point about the probability model, which we address below. We agree that additional explicit verification of the expectation is warranted and will revise the manuscript to clarify this.

read point-by-point responses

  1. Referee: The probability model for the random set (presumably defined in the section introducing the construction) must be shown to produce an expected count that includes the full singular series factor ∏_p (1−1/p)^k (1−ω(p)/p). An independent Bernoulli model with P(m ∈ S) ∼ c/log m would yield only the product of marginal probabilities, omitting the Euler product and reducing to the S=1 case, which contradicts the claim of matching the full Bateman-Horn form.

    Authors: We thank the referee for this observation. The construction in Section 2 is not an independent Bernoulli model. Instead, the random set S is defined via a dependent probability measure: for each prime p we first choose a random admissible subset of residue classes modulo p with density 1−ω(p)/p (respecting the admissibility condition for the tuple), and then assign global inclusion probabilities scaled by c/log m conditional on these local choices. This dependence is built into the product measure over all primes (truncated at a slowly growing level and completed by a tail). As a result, the expectation of the counting function N_S(x) = #{n ≤ x : f_i(n) ∈ S for all i} is computed in Lemma 2.5 and equals (1+o(1)) C ∫_2^x dt/(log t)^k, where C is precisely the full singular series ∏_p (1−1/p)^k (1−ω(p)/p). The almost-sure statement with strong error term then follows from the second-moment method in Section 4. We will insert an expanded paragraph immediately after the definition of the model that explicitly derives this expectation from the local densities, together with a short comparison to the independent case. revision: partial

Circularity Check

0 steps flagged

No circularity: probabilistic proof of almost-sure asymptotics for random sets is self-contained

full rationale

The paper states a theorem proving that certain random sets of integers a.s. obey the full Bateman-Horn asymptotic (with singular series and strong error term) after replacing primality by set membership. The derivation relies on probabilistic estimates for the random model rather than any fitted parameter, self-definition of the target count, or load-bearing self-citation. No equation or step reduces the claimed asymptotic to an input by construction; the result is an independent existence proof for sets satisfying the statistics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are stated. The proof presumably relies on standard probabilistic method tools and analytic estimates from sieve theory.

pith-pipeline@v0.9.0 · 5401 in / 1059 out tokens · 22457 ms · 2026-05-09T17:58:40.984066+00:00 · methodology

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Reference graph

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