arxiv: 2512.01739 · v2 · submitted 2025-12-01 · 🧮 math.NT

Recognition: no theorem link

Quantitative correlations and some problems on prime factors of consecutive integers

Terence Tao , Joni Ter\"av\"ainen

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Pith reviewed 2026-05-17 02:36 UTC · model grok-4.3

classification 🧮 math.NT

keywords prime divisorsconsecutive integersomega functionirrational seriesasymptotic countsmultiplicative functionsprobabilistic methodErdős conjectures

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The pith

There are infinitely many n such that omega(n+k) is at most a constant times k for every k, the series sum omega(n)/2^n is irrational, and the count of n up to x with omega(n) equal to omega(n+1) obeys an asymptotic for almost all x.

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

The paper establishes three results on the prime-divisor functions omega, Omega, and tau. It proves there are infinitely many starting integers n where, no matter how far one looks ahead by any fixed k, both the distinct and total prime factors of n+k remain bounded by a multiple of k. It shows that summing omega(n) divided by successive powers of two produces an irrational number. It also derives an asymptotic formula, valid at almost every scale x, for how often consecutive integers n and n+1 share exactly the same number of distinct prime factors, with parallel statements for the total prime count and the divisor function. These statements resolve long-standing conjectures of Erdős, Straus, Pomerance, and Sárközy. The arguments rest on the probabilistic method, a high-dimensional sieve in one case, and a quantitative correlation estimate for multiplicative functions in the others.

Core claim

The authors prove that there exist infinitely many positive integers n such that omega(n+k) ≤ Omega(n+k) ≪ k for every positive integer k, confirming a conjecture of Erdős and Straus. They establish the irrationality of the series sum omega(n) 2^{-n}, settling a conjecture of Erdős. They obtain an asymptotic formula for the number of n ≤ x with omega(n) = omega(n+1) that holds for almost all x, together with analogous formulas for Omega and tau, as conjectured by Erdős, Pomerance, and Sárközy. All three conclusions are reached by the probabilistic method, using a Maynard-type high-dimensional sieve for the first result and a general quantitative two-point correlation estimate for the second.

What carries the argument

A quantitative two-point correlation estimate for multiplicative functions that saves a small power of the logarithm, which controls the joint distribution of omega and Omega at nearby arguments.

If this is right

The Erdős-Straus conjecture on the existence of n with omega(n+k) ≪ k for all k is settled.
The Erdős conjecture asserting irrationality of sum omega(n) 2^{-n} is settled.
The Erdős-Pomerance-Sárközy conjecture on the asymptotic count of n with omega(n) = omega(n+1) holds for almost every x.
Parallel asymptotic formulas hold when omega is replaced by Omega or by the divisor function tau.

Where Pith is reading between the lines

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

The same correlation machinery could be used to study the joint distribution of omega over longer arithmetic progressions or short intervals.
The irrationality result suggests that many other naturally occurring series built from omega or Omega are likewise irrational.
The existence result for bounded prime factors may extend to show that omega remains small on all terms of certain polynomial sequences simultaneously.

Load-bearing premise

The quantitative two-point correlation estimate for multiplicative functions holds with enough uniformity across the needed ranges of parameters to reach the irrationality and asymptotic-count applications.

What would settle it

A direct numerical check, for a sequence of large x, showing that the proportion of n ≤ x with omega(n) = omega(n+1) deviates from the predicted main term by more than the error allowed by the asymptotic.

Figures

Figures reproduced from arXiv: 2512.01739 by Joni Ter\"av\"ainen, Terence Tao.

**Figure 1.** Figure 1: A schematic diagram (not to scale) of the various ranges of primes for a given shift. We do not perform any subdivision of primes when k is a very far shift. We perform an exact sieve at tiny primes and a Selberg type sieve at medium primes, in order to ensure ω(n + k) ≪C0 k with high probability for all shifts k. From the crude bound Ω(n) ≪ log n we have Ω(n+k) ≪ k for all very far shifts k > K+, so (2.1)… view at source ↗

**Figure 2.** Figure 2: The probability of the event that τ (n + 1)/τ (n) is a power of 2 numerically converges (fairly rapidly) to about 0.4888. For comparison we also include the larger events νp(τ (n + 1)/τ (n)) = 0 for p = 3, p = 3, 5, and p = 3, 5, 7, which are easier to calculate asymptotically (they correspond to return probabilities of certain one-dimensional, two-dimensional, and three-dimensional random walks respecti… view at source ↗

**Figure 3.** Figure 3: The probabilities P(f(n) = f(n + 1)) for f = ω, Ω, τ and 10 ≤ n ≤ 107 , normalized by multiplying against 2p π(log2 x + B5), 2p π(log2 x + B6), and 2 p π log2 x/cτ respectively, with the x-axis plotted using log2 x. (Due to the negativity of B5, the normalized f = ω probability vanishes for small x.) Theorem 1.6 asserts that these densities converge to 1 asymptotically, although the convergence is very sl… view at source ↗

**Figure 4.** Figure 4: Empirical values of B5, B6, B7 imputed from treating the heuristic approximations (1.8), (1.9), (1.13) as exact, compared against the predicted values of B5 and B6. These numerics tentatively suggest that B7 ≈ −1 to one significant figure [PITH_FULL_IMAGE:figures/full_fig_p057_4.png] view at source ↗

**Figure 5.** Figure 5: Histograms of ω(n+1)−ω(n) and Ω(n+1)−Ω(n) for n ≤ 107 , compared to the Gaussian with the empirical mean ˆµ and variance ˆσ 2 for these data sets, as well as the predicted Gaussian behavior using (1.8), (1.9). At this scale the effects of the lower order terms B5, B6 are significant [PITH_FULL_IMAGE:figures/full_fig_p058_5.png] view at source ↗

**Figure 6.** Figure 6: Histogram of log τ(n+1)−log τ(n) log 2 for those n ≤ 107 with τ (n + 1)/τ (n) a power of two, compared to the Gaussian with the empirical mean ˆµ and variance ˆσ 2 for this data set, as well as the predicted Gaussian behavior setting B7 = 0. There is a significant deviation, suggesting that the undetermined constant B7 is negative. References [1] H. Andr´es H. and M. Radziwi l l. Expansion, divisibility an… view at source ↗

read the original abstract

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that $\omega(n+k) \leq \Omega(n+k) \ll k$ for all positive integers $k$, establishing a conjecture of Erd\H{o}s and Straus. Secondly, we show that the series $\sum_{n=1}^{\infty} \omega(n)/2^n$ is irrational, settling a conjecture of Erd\H{o}s. Thirdly, we prove an asymptotic formula conjectured by Erd\H{o}s, Pomerance and S\'ark\"ozy for the number of $n\leq x$ satisfying $\omega(n)=\omega(n+1)$, for almost all $x$, with similar results for $\Omega$ and $\tau$. Common to the resolution of all these problems is the use of the probabilistic method. For the first problem, this is combined with computations involving a high-dimensional sieve of Maynard-type. For the second and third problems, we instead make use of a general quantitative estimate for two-point correlations of multiplicative functions with a small power of logarithm saving that may be of independent interest. This correlation estimate is derived by using recent work of Pilatte.

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

Tao and Teräväinen resolve three Erdős-era conjectures on prime factors of consecutive integers via probabilistic methods and a Pilatte-derived correlation bound.

read the letter

Tao and Teräväinen resolve three longstanding conjectures on the distribution of ω, Ω, and τ for consecutive integers. They establish infinitely many n where ω(n+k) ≤ Ω(n+k) ≪ k holds for all positive k, prove that ∑ ω(n)/2^n is irrational, and give an asymptotic for the count of n ≤ x with ω(n)=ω(n+1) that holds for almost all x, with matching statements for Ω and τ. All three rest on the probabilistic method, with a Maynard-type high-dimensional sieve for the first and a quantitative two-point correlation estimate for multiplicative functions for the second and third. The correlation bound, which saves a small power of logarithm and is extracted from Pilatte's recent work, is presented as potentially useful on its own. The paper does a clean job showing how the same tools handle the different problems and makes the logical steps explicit. The first result benefits from the sieve machinery being already well-developed, while the correlation estimate supplies the needed control for the generating-function discrepancy in the irrationality claim and for showing the exceptional set has density zero in the almost-all asymptotic. The main soft spot is whether the uniformity and saving in that correlation lemma are strong enough to absorb the error terms that arise when it is inserted into the two later arguments; any shortfall in the parameter range would affect both conclusions. The paper should lay out the explicit ranges and error bookkeeping so that this can be checked without extra work. This is for analytic number theorists who follow multiplicative functions, short-interval statistics, and probabilistic methods in number theory. Readers interested in Erdős-type problems or recent sieve and correlation techniques will get direct value. It deserves a serious referee because the claims are precise, the methods are current, and the results organize several related phenomena at once. I would send it to peer review.

Referee Report

1 major / 3 minor

Summary. The manuscript presents resolutions to three problems on the prime divisor functions of consecutive integers. It proves that there are infinitely many positive integers n satisfying ω(n+k) ≤ Ω(n+k) ≪ k for all positive integers k, confirming a conjecture of Erdős and Straus. It establishes the irrationality of the infinite series ∑_{n=1}^∞ ω(n)/2^n, addressing a conjecture of Erdős. Additionally, it provides an asymptotic formula for the count of n ≤ x with ω(n) = ω(n+1) that holds for almost all x, with analogous results for Ω and τ, as conjectured by Erdős, Pomerance, and Sárközy. The common methodology involves the probabilistic method, with a high-dimensional Maynard-type sieve for the first result and a quantitative two-point correlation estimate for multiplicative functions, derived from Pilatte's recent work, for the second and third results.

Significance. Assuming the technical details are verified, these results are significant as they resolve longstanding conjectures in multiplicative number theory. The proofs highlight the utility of probabilistic methods in conjunction with modern sieve techniques and correlation estimates. The general correlation lemma may be useful in other contexts involving multiplicative functions and could stimulate further research. The paper appropriately credits the foundational ideas from Erdős and the recent contributions of Pilatte.

major comments (1)

[§3, Lemma 3.1] §3, Lemma 3.1 (two-point correlation estimate): The quantitative bound with small logarithmic saving, derived from Pilatte, is load-bearing for Theorems 2 and 3. The manuscript must explicitly verify that the uniformity range and saving exponent suffice to absorb error terms when applied to the discrepancy control in the irrationality proof (§5) and to show the exceptional set has density zero in the almost-all asymptotic (§6); without this parameter check the applications do not go through.

minor comments (3)

The abstract would be clearer if it briefly indicated which method applies to which of the three results.
[Introduction] Notation for ω, Ω, and τ is standard but a short reminder in the introduction would aid readers.
[References] Ensure the full reference to Pilatte's paper includes the precise theorem or result invoked for the correlation estimate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of the significance of the results. We address the major comment below and will make the requested changes in the revised version.

read point-by-point responses

Referee: [§3, Lemma 3.1] §3, Lemma 3.1 (two-point correlation estimate): The quantitative bound with small logarithmic saving, derived from Pilatte, is load-bearing for Theorems 2 and 3. The manuscript must explicitly verify that the uniformity range and saving exponent suffice to absorb error terms when applied to the discrepancy control in the irrationality proof (§5) and to show the exceptional set has density zero in the almost-all asymptotic (§6); without this parameter check the applications do not go through.

Authors: We agree that an explicit verification of the parameters is required for the applications to be fully rigorous. In the revised manuscript we will insert a dedicated parameter-check subsection at the end of §5 and a corresponding paragraph in §6. For the irrationality result, the discrepancy control in the partial sums of ω(n)/2^n relies on the two-point correlation bound holding uniformly for |h| ≤ (log x)^C with a saving of (log log log x)^δ for some δ>0; we will verify that the saving exponent inherited from Pilatte’s work is large enough to dominate the O(1/log log x) error arising from the truncation and the contribution of the exceptional set. For the almost-all asymptotic, the same correlation estimate is applied with h=1 and x ranging over a short interval; we will confirm that the uniformity range (which extends to shifts h ≪ x^θ for θ<1) together with the logarithmic saving suffices to make the measure of the exceptional set o(x). These checks follow directly from the stated range in Lemma 3.1 and the error terms already present in §§5–6; we will spell them out with explicit numerical constants for the exponents. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results derived from external Pilatte correlation estimates and probabilistic method

full rationale

The paper proves its three main claims by combining the probabilistic method with a quantitative two-point correlation bound for multiplicative functions (derived from Pilatte's independent recent work) and, for the first claim, a Maynard-type high-dimensional sieve. No derivation step reduces a claimed prediction or theorem to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the correlation estimate is invoked as an external input whose uniformity is taken as given for the applications. The central results therefore remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard axioms of analytic number theory such as properties of multiplicative functions and sieve theory; no free parameters or invented entities introduced in the abstract.

axioms (1)

domain assumption Standard properties of the functions ω, Ω, τ and multiplicative functions in short intervals
Invoked throughout the applications of the probabilistic method and correlation estimates.

pith-pipeline@v0.9.0 · 5534 in / 1213 out tokens · 62233 ms · 2026-05-17T02:36:49.634372+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Improved bounds for the Fourier uniformity conjecture
math.NT 2026-04 conditional novelty 6.0

The summed supremum of short-interval Fourier transforms of λ(n) is o(HX) for H ≥ exp((log X)^{2/5+ε}).
On the Number of Prime Factors of Consecutive Integers
math.NT 2026-04 unverdicted novelty 6.0

There are infinitely many n such that ω(n+k) ≪ log k for all k ≥ 2.

Reference graph

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