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Transforming the Erd\H{o}s-Kac theorem

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper transforms the Erdős–Kac theorem to build interval estimates for the prime-divisor count, with a square-root variance stabilizer, a three-quarters width optimizer, and a trained score interval for small integers.

desk verdict The delta-method/Box-Cox part of this paper is correct and genuinely useful, but the trained-interval reliability claims sit on a calibration that the paper never explains. read the letter →

arxiv 2506.08503 v1 pith:U2GZED6T submitted 2025-06-10 math.NT math.PRmath.STstat.TH

Kimihiro Noguchi This is my paper
classification math.NTmath.PRmath.STstat.TH MSC 11N4062E2060F05
keywords Box-CoxtransformationfuzzyintervalestimatePoissonscorevariance-stabilizingErdős–Kactheoremprimeomegafunctiondeltamethod
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends the Erdős–Kac theorem, which says that the number $\omega(m)$ of distinct prime divisors of $m$ is asymptotically normal with mean and variance $\log\log m$, by showing that the theorem survives a nonlinear change of scale. Provided a transformation $g$ is differentiable at $1$ with nonzero derivative, the ratio $g(\omega/\ell_2)$ is also asymptotically standard normal after the usual centering and scaling, where $\ell_2(m)=\log\log m$. Applying the Box-Cox family of transformations singles out two useful members: the square root, which makes the limiting variance a constant independent of $m$, and the three-quarters power, which asymptotically minimizes the width of a two-sided interval estimate for $\omega$. The paper then constructs a score-type interval estimate from the transformed theorem and evaluates all these intervals for small integers using fuzzy coverage probabilities, finding that a Poisson-based interval is the most reliable without training and that the score and Poisson intervals become accurate after a simple training step. If these claims hold, statisticians and number theorists get practical, computable interval estimates for how many distinct prime factors an integer has, including for numbers far smaller than the regime where the original theorem is usually trusted.

What carries the argument

The argument is carried by the delta method fitted into the probabilistic framework of additive arithmetic functions. Starting from the central limit theorem for additive functions, the paper introduces the remainder function $r(x) = (g(x)-g(1))/(x-1) - g'(1)$, which is continuous at $1$, and uses Slutsky's theorem under the discrete uniform measure to prove that $g(f_n/A_n)$ is asymptotically normal whenever the additive function $f_n$ is (Theorem 2.2). Specializing $f_n=\omega$, $A_n=\ell_2 + O(1)$, $B_n^2=\ell_2 + O(1)$ gives Theorem 1.1. The Box-Cox transformation $y_\lambda(x) = (x^\lambda-1)/\lambda$ (or $\log x$ at $\lambda=0$) turns this into an explicit $\lambda$-family of normal approximations; the asymptotic expansion of the interval width, whose first correction term contains the factor $(\lambda-1)(2\lambda-1)$, identifies $\lambda=3/4$ as the width-minimizer. For small integers, the paper replaces ordinary coverage probability with a fuzzy coverage probability that partially credits the fractional upper and lower endpoints of an interval, and it estimates local adjustment functions $f_{\mu,\lambda}$ and $f_{\sigma,\lambda}$ by fitting power functions of linear functions of $\ell_2$ to smoothed $\omega$. The score interval estimate comes from solving $(\omega-\ell_2)/\sqrt{\omega} = \pm z$ for $\omega$, in the manner of Wilson's interval for a binomial proportion.

What would settle it

Take the actual values of $\omega(m)$ for $m$ near $10^8$ and $10^{12}$, compute the smoothed mean and standard deviation of $\omega^\lambda$ with the same window and powers as Section 7, and compare them with the fitted functions in Table 5; a deviation of more than a few percent would show that the power-of-a-linear-function model does not extrapolate, and the trained interval estimates' claimed out-of-sample reliability would fail. An even more direct check is to compute the exact fuzzy coverage probabilities of the trained score and Poisson intervals at $m=10^{13}$ and $m=10^{14}$ and see whether they remain inside Bradley's liberal band around the nominal $0.6319$.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.1: if $g$ is differentiable at $1$ and $g'(1)\neq 0$, then under the discrete uniform measure on $\{1,\dots,n\}$, the transformed ratio $(g(\omega/\ell_2)-g(1))/(g'(1)/\sqrt{\ell_2})$ converges in distribution to the standard normal $\Phi$. With $g$ equal to the Box-Cox power $y_\lambda$, this becomes $(\omega^\lambda - \ell_2^\lambda)/(\lambda \ell_2^{\lambda-1/2}) \Rightarrow \Phi$, and the same argument gives versions with local adjustment functions $f_\mu$ and $f_\sigma$ in place of $\ell_2$ and $\sqrt{\ell_2}$. The two distinguished powers are $\lambda=1/2$, where the denominator no longer depends on $m$ so the variance is stabilized, and $\lambda=3/4$, which minimizes the asymptotic width of the two-sided interval because the first nonconstant term in the width's expansion is proportional to $(\lambda-1)(2\lambda-1)$. Solving the quadratic obtained by standardizing with $\omega$ in the denominator gives a score interval estimate in the spirit of Wilson, and numerical work with fuzzy coverage probabilities shows that the score interval, and the Poisson interval based on Landau's formula, keep their coverage closest to nominal for $m$ between $10^5$ and $10^{14}$ once their means and standard deviations are estimated from $\omega$ values on $[10^4,10^6]$. The same transformation machinery is applied to the Erdős–Pomerance theorem, where it yields variance stabilization at $\lambda=1/4$ while the optimal width remains at $\lambda=3/4$.

Load-bearing premise

The load-bearing premise outside the asymptotic theory is the Section 7 working assumption that the mean and standard deviation of the transformed prime-divisor count can be represented as fixed powers of linear functions of $\log\log m$, with coefficients fitted on $m\in[10^4,10^6]$ and trusted to extrapolate to $m=10^{14}$ and beyond; if that empirical model drifts, the claimed reliability of the trained score and Poisson intervals for out-of-sample $m$ does not follow.

Editorial extensions

If this is right

  • The square-root transformation yields a simple rarity rule: integers with $|\sqrt{\omega(m)} - \sqrt{\log\log m}| > 1.5$ make up roughly 1 in 400 of all integers, and >2.0 roughly 1 in 16,000, giving an easily remembered scale for interpreting $\omega$.
  • The three-quarters power gives the asymptotically narrowest two-sided $100(1-\alpha)\%$ interval for $\omega$ near large $m$: $[\ell_2(1 - 3z/(4\sqrt{\ell_2}))^{4/3}, \ell_2(1 + 3z/(4\sqrt{\ell_2}))^{4/3}]$, a direct improvement over the usual $\ell_2 \pm z\sqrt{\ell_2}$ interval.
  • The score interval estimate of $\omega$, obtained by solving the quadratic $(\omega-\ell_2)^2/\omega = z^2$, is the most reliable of the normal-based intervals after training, with fuzzy coverage probabilities close to nominal for $m \in [10^5, 10^{14}]$.
  • The Poisson interval estimate, centered at $\ell_2(m)+1$ with width calibrated to the nominal level, is relatively reliable even without training, and it tends to be narrower than the transformed Erdős–Kac intervals (e.g., at $m\approx 10^{70}$ it gives $[4.52, 7.65]$ versus Billingsley's $[3.05, 7.11]$).
  • The transformation results transfer to the Erdős–Pomerance theorem: $\omega(\varphi(m))$ is normal after a Box-Cox-type transform with variance stabilization at $\lambda=1/4$ and asymptotically optimal interval width at $\lambda=3/4$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because variance stabilization at $\lambda=1/2$ makes deviations comparable across different $m$, one could define a universal rarity score $z = (\sqrt{\omega} - \sqrt{\ell_2})/0.5$ and rank integers of vastly different sizes on one scale; the paper does not propose such a score but its own Table 1 is the seed of it.
  • The training procedure is only validated on $m$ up to $10^{14}$; a natural robustness test is to retrain on shifted windows (e.g., $10^6$–$10^8$) and see whether the fitted power-law forms in Table 5 drift, which would indicate that the choice of training range, rather than the asymptotic theory, drives the reported coverage.
  • The relative success of the Poisson interval estimate supports a shifted-Poisson model as a better finite-sample description of $\omega$ than the normal law; the paper's discussion of Landau's formula and the shifted-Poisson mass function marks the Poisson approximation as a natural target for a rate-of-convergence theorem.
  • The fuzzy-coverage device is transferable: any interval for a lattice-valued statistic whose endpoints fall between integers suffers the same jump problem, so the same fractional-credit definition could be applied to binomial, Poisson, or hypergeometric intervals, not just to $\omega$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper applies the delta method to Billingsley's probabilistic version of the Erdős–Kac theorem and obtains a central limit theorem for g(ω/ℓ2) when g'(1)≠0. Choosing g as a Box–Cox power gives a transformed Erdős–Kac theorem; the paper identifies λ=1/2 as the variance-stabilizing transformation and λ=3/4 as the power that asymptotically minimizes the width of the two-sided interval estimate for ω. It then introduces local adjustment functions to refine the intervals for finite m, develops fuzzy coverage probabilities to handle the discreteness of ω, and compares five interval estimates (λ=1/2, 3/4, 1, Poisson, and score) for m∈[10^5,10^14]. After fitting mean and standard deviation functions on m∈[10^4,10^6], it claims that the trained score and Poisson intervals are reliable. The paper also states analogous transformed Erdős–Pomerance theorems.

Significance. If the main claims hold, the theoretical part is a clean and useful contribution: it gives a principled family of Erdős–Kac-based intervals, and the variance-stabilization and width-optimality results are natural and well-motivated. The asymptotic expansion behind Theorem 4.1 is correct, and the fuzzy-coverage framework is an appropriate response to the discreteness of ω. The claimed numerical advantage of the trained score and Poisson intervals, however, is not supported by the calibration evidence as presented: the fitted standard deviations in Table 5 are markedly smaller than the Erdős–Kac variance even inside the training range, so the reliability conclusion rests on unverified numerics. The paper would also be strengthened by providing the omitted proofs and by making the numerical evaluation reproducible.

major comments (3)
  1. [Section 7, Table 5, Figure 8] The fitted scale parameters in Table 5 appear inconsistent with the Erdős–Kac variance on the training range. At m=10^5 (ℓ2≈2.44), the λ=1 value gives f̂σ,1≈0.0499+0.3677·2.44≈0.95, whereas the Erdős–Kac variance of ω at that scale is ℓ2≈2.44, so the empirical standard deviation should be about 1.56, not 0.95. For λ=1/2, the fitted value is about 0.60 instead of the variance-stabilized value 1. This is not a small finite-sample correction. Since the trained interval widths in Figure 8 are built from these f̂σ,λ values, the reported reliability of the trained Box-Cox, score, and Poisson intervals is not established. Please reconcile Table 5 with a direct computation of the residual standard deviations on the training range, or supply the code/data so that the reader can verify the calibration.
  2. [Theorems 3.1, 8.1–8.3] Theorem 3.1 is the central result used for all subsequent interval constructions, and Theorems 8.1–8.3 are stated in Section 8, but their proofs are omitted 'for brevity.' At least for Theorem 3.1, a short derivation from Theorem 1.1 should be included; for Section 8, either provide the analogous delta-method derivations or state explicitly that they follow from (25) by the same argument. As written, the reader cannot verify the λ=0 cases or the rates without reconstructing the calculations.
  3. [Theorem 8.1 and Theorem 8.2] The displayed denominator in Theorem 8.1, 2g'(1)/(√3 ℓ2), is not consistent with Theorem 8.2 or with (25). Substituting g(x)=yλ(x) into that denominator and into the numerator gives a ratio with a denominator containing ℓ2^{2λ−1}, not ℓ2^{2λ−1/2} as in Theorem 8.2; the λ=1 case would not reduce to (25). The correct denominator should be 2g'(1)/(√3√ℓ2). The same notational correction is needed in the λ=0 statement of Theorem 8.2 and in the following sentence, so that the variance-stabilizing power λ=1/4 is derived from the correct rate.
minor comments (4)
  1. [Section 1] There are several typographical slips, including 'the the' in the introduction and 'the the vicinity' in Section 4; a careful proofreading pass is needed.
  2. [Section 6] The sentence beginning 'when ⌈Lλ,α(m)⌉=⌊Uλ,α(m)⌋+1, we may further assume that at least one of ... is included' is not reflected in the fuzzy coverage formula that follows; the intended convention should be stated precisely or removed.
  3. [Section 7] The numerical results in Tables 4–5 and Figures 4–8 are not reproducible from the text alone because no code or detailed data-processing pipeline is given; providing the code or a detailed pseudocode would substantially help the reader check the calibration issue raised above.
  4. [Section 9] The claim that 'all the theoretical results follow even if we replace ω with Ω' is stated without comment; since Ω is also an additive function satisfying the relevant Billingsley conditions, a one-sentence justification would make the remark self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central CLT and optimization results derive from external theorems (Billingsley, Mertens, Hardy-Ramanujan, Landau, Erdős-Pomerance); the Section 7 training is empirical and transparently evaluated.

full rationale

The derivation chain starts from Billingsley's Erdős-Kac theorem and Mertens' estimates, then applies standard delta-method/Slutsky arguments (Theorems 2.2, 1.1, 3.1). The variance-stabilizing choice λ=1/2 and the width-optimal λ=3/4 are obtained by algebraically minimizing the asymptotic interval width derived from Theorem 3.1, not by fitting to data. The score interval is obtained by inverting the asymptotic normal pivot in Corollary 3.2.2, following Politis (2024), an external source. The Poisson interval is calibrated to the Landau asymptotic formula (16), and its performance is then measured by fuzzy coverage probabilities computed from data; this measurement is not forced by the calibration. The Section 7 training fits f̂_{μ,λ} and f̂_{σ,λ} to m∈[10^4,10^6] and then evaluates fuzzy coverage for m∈[10^5,10^14]; although some evaluation points lie inside the training range, the paper explicitly distinguishes in-sample from out-of-sample performance, and the fitted parameters do not directly set the empirical coverage to the nominal level. The only self-citation (Noguchi and Ward 2024) is a supporting reference for the standard square-root variance-stabilizing property and plays no load-bearing role. Concerns that the fitted standard deviations are miscalibrated relative to the Erdős-Kac variance are empirical correctness risks, not circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The central theorems rest on standard probabilistic number theory (Billingsley, Mertens, Hardy-Ramanujan) plus empirical fits in Section 7. No new unverifiable physical entities are introduced; the local adjustment functions are data-driven and thus carry an independent handle.

free parameters (6)
  • beta_{0,lambda}, beta_{1,lambda} (mean fit) = lambda=1/2: (0.4300, 0.9152); lambda=3/4: (0.4284, 0.9343); lambda=1: (0.4270, 0.9527)
    Fitted by nonlinear least squares to smoothed omega^lambda on m in [10^4,10^6] in Step 3 of Section 7 (Table 5).
  • gamma_{0,lambda}, gamma_{1,lambda} (standard deviation fit) = lambda=1/2: (-0.0109, 0.1498); lambda=3/4: (0.0196, 0.2695); lambda=1: (0.0499, 0.3677)
    Fitted by nonlinear least squares to smoothed standard deviations in Step 6 of Section 7 (Table 5).
  • eta_0, eta_1 (trained score interval variance fit) = (-0.1136, 0.3855)
    Least squares fit of sigma_1^2 against omega_1 in Section 7, used to construct the trained score interval estimate.
  • Window j and grid increment = j=2000, increment=1000, training range [10^4,10^6]
    Hand-chosen in Section 7 to reduce computational burden; affects the smoothed mean and standard deviation estimates.
  • Power selection q_mu, q_sigma = q_mu=1, q_sigma=1/lambda
    Chosen by maximizing sample correlation in Section 7; these choices determine the parametric form of the fitted mean and SD.
  • kappa_alpha(m) (Poisson interval half-width) = varies with m
    Set so that the asymptotic fuzzy coverage probability equals 1-alpha for the Poisson interval estimate (Equation 20).
assumptions (6)
  • domain assumption Billingsley's Theorem 2.1 and Theorem 3.1 on additive functions
    Used as the base central limit theorem in Lemma 2.1 and Theorem 2.2; cited from Billingsley (1974).
  • standard math Mertens' second theorem
    Used in Section 2 to identify A_n and B_n^2 with ell_2(n) + O(1) for the prime omega function.
  • domain assumption Hardy-Ramanujan theorem on the normal order of omega
    Used to show omega/ell_2 converges to 1 in probability, which is needed for the delta method and Corollary 3.2.2.
  • domain assumption Landau's asymptotic formula for rho_d(m)
    Used in Equation (16) to compute asymptotic coverage probabilities and to define the Poisson interval estimate.
  • domain assumption Erdős-Pomerance theorem
    Used as the base result for the transformed results in Section 8.
  • standard math Slutsky's theorem and the delta method under P_n
    Used throughout the proofs of the main theorems; the paper notes these hold under the discrete uniform measure as n tends to infinity.
invented entities (1)
  • Local adjustment functions f_mu,lambda and f_sigma,lambda independent evidence
    purpose: To refine the mean and standard deviation in the transformed Erdős-Kac theorem for finite m, allowing the trained interval estimates in Section 7.
    The functions are estimated from omega data on [10^4,10^6] and validated against out-of-sample fuzzy coverage probabilities, so they have an empirical falsifiable handle.

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Cite this review

Pith. "Pith review of Transforming the Erd\H{o}s-Kac theorem." pith.science (2026). https://pith.science/paper/U2GZED6T

@misc{pith2026250608503,
  author       = {Pith},
  title        = {Pith review of: Transforming the Erd\Hos-Kac theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U2GZED6T}},
  note         = {Machine review of arXiv:2506.08503}
}
read the original abstract

Transforming the Erd\H{o}s-Kac theorem provides more flexibility in how the theorem can be utilized as an interval estimate for the prime omega function, which counts the number of distinct prime divisors. Here, we consider a direct transformation by the delta method. Then, we demonstrate that the square-root and three-quarters power asymptotically achieve variance stabilization and an optimal width, respectively. Furthermore, by adjusting the denominator of the theorem, we derive the score interval estimate. To make these interval estimates reliable for small positive integers, we examine performances of various interval estimates for the prime omega function using fuzzy coverage probabilities. The results indicate that the score interval estimate performs well even for small positive integers after training the mean and standard deviation using the prime omega function. Moreover, the Poisson interval estimate is relatively reliable with or without training. Additional theoretical results on the Erd\H{o}s-Pomerance theorem are also provided.

Figures

Figures reproduced from arXiv: 2506.08503 by the authors.

Figure 1
Figure 1. The optimal value of λ as a function of ℓ2(m) at zα/2 = 2 (left) and at zα/2 = 3 (right). The horizontal dashed line is drawn at λ = 3/4. The vertical dashed line represents the point at which the lower bound exceeds 1. To give a sense of how the choice of zα/2 affects L −1 λ,α(1), [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Plot of pm,j,1,α⋆ with j = 2000 against m for m ∈ [4 · 104 , 5 · 105 ]. The first three vertical lines indicate U −1 1,α⋆ (1) − j, U −1 1,α⋆ (1), and U −1 1,α⋆ (1) + j, and the last three vertical lines indicate L −1 1,α⋆ (4) − j, L −1 1,α⋆ (4), and L −1 1,α⋆ (4) + j. Now, let us consider log(si+1,α/si,α), i ∈ N, which measures the separation between the i-th and (i + 1)-th occurrence of m such that ω(m) exceeds U1,… view at source ↗
Figure 3
Figure 3. 3D Plot of the log(si+1,α/si,α) (Separation) against i (Index) with zα/2 ∈ [0.5, 1.5] (Threshold). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Plot of fuzzy coverage probabilities for the five interval estimates of [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: Plot of fuzzy coverage probabilities for the three adjusted interval estimates of [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]
Figure 6
Figure 6. Figure 6: Plots of smoothed means [˜ωλ(m)]1/λ against ℓ2(m), m ∈ [104 , 106 ], for λ ∈ {1/2, 3/4, 1}. Let ˆfµ,λ = βˆ 0,λ + βˆ 1,λℓ2. In the next step, the standard deviation of (ω λ − ˆf λ µ,λ)/λ is estimated by σ˜λ(m) =   1 2j + 1 mX +j t=m−j [PITH_FULL_IMAGE:figures/full_fi…
Figure 7
Figure 7. Figure 7: Plots of smoothed standard deviations (SDs) ˜σ [PITH_FULL_IMAGE:figures/full_fig_p033_7.png]
Figure 8
Figure 8. Figure 8: Plot of fuzzy coverage probabilities of the five trained interval estimates of [PITH_FULL_IMAGE:figures/full_fig_p036_8.png]

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