A subset generalization of the Erd\H{o}s-Kac theorem over number fields with applications

Sourabhashis Das; Wentang Kuo; Yu-Ru Liu

REVIEW 2 major objections 3 minor 19 references

A subset generalization of the Erd\H{o}s-Kac theorem over number fields with applications

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A subset generalization of the Erdős–Kac theorem: over a number field, any ideal subset meeting the paper's uniformity conditions has its distinct-prime-ideal count $\omega(m)$ converge to a Gaussian distribution after normalization, and…

desk verdict Useful subset Erdős–Kac framework, but the h-full application has an inverted local factor that invalidates Theorem 1.4 as written. read the letter →

arxiv 2506.03215 v1 pith:TRRENYMO submitted 2025-06-03 math.NT

Sourabhashis Das , Wentang Kuo , Yu-Ru Liu This is my paper

classification math.NT MSC 11R0411R4411R45

keywords numberfieldsprimeidealsomega-functionErdős-Kactheoremh-freeh-fullGaussiandistributionidealcounting

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

The paper proves a conditional subset-level Erdős–Kac theorem: for any collection $S$ of ideals in a number field whose counting function is not concentrated on norms below $\sqrt{x}$ and whose divisibility events are approximately independent with controlled errors, the number $\omega(m)$ of distinct prime ideal divisors follows the standard Gaussian distribution after subtracting $\log\log N(m)$ and dividing by $\sqrt{\log\log N(m)}$. The classical law covers all integers, while many sparse structured families are outside its reach. The paper shows that the $h$-free ideals and the $h$-full ideals of a number field satisfy the regularity conditions, so the Gaussian law holds over those families as well. The proof is a moment-matching argument: it compares $\omega(m)$ with a sum of independent Bernoulli random variables indexed by prime ideals.

What carries the argument

The central object is the truncated independent sum $S_y=\sum_{N(\wp)\le y}X_\wp$, where the $X_\wp$ are independent Bernoulli variables with $P(X_\wp=1)=\lambda_\wp$, and $\lambda_\wp$ is the asymptotic density of $S$-elements divisible by $\wp$. Lemmas 3.1--3.5 show that the normalized moments of the truncated divisor-count $\omega_y(m)$ match those of $S_y$ as $x\to\infty$, provided conditions (d), (e), and especially (f) control the errors. The Gaussian law for $S$ is thereby reduced to the classical central limit theorem for independent sums (Fact 5). For the applications, the counting functions of $h$-free and $h$-full ideals are factored into Euler products, and the co-prime-to-$\ell$ subcounts (Lemmas 5.1 and 6.2) give the densities $\lambda_\wp$ and the joint-error estimates needed for condition (f).

What would settle it

Take $K=\mathbb{Q}$, $h=2$ (so $h$-free ideals are squarefree integers), and compare the exact count of $n\le x$ divisible by two distinct primes $p,q$ with the formula claimed after (14). The asserted error is $O(1/(x^{1-\tau}(pq)^\tau))$; computing this error for many pairs $p,q$ near $y=x^{1/\log\log x}$ and checking whether the sum over all such pairs is $o((\log\log x)^{-r/2})$ for each $r$ would settle whether condition (f) really holds for the $h$-free family.

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Extended reading notes

Core claim

On its own terms, the central claim is Theorem 1.1: if $S$ is an infinite subset of ideals of $\mathcal{O}_K$ with $|S(x^{1/2})|=o(|S(x)|)$, and there exists $\beta\in(0,1]$ and $y=y(x)<x^\beta$ such that conditions (a)--(f) hold, then for every real $\gamma$, $$\lim_{x\to\infty}\frac{1}{|S(x)|}\left|\left\{m\in S(x):N(m)\ge 3,\ \frac{\omega(m)-\log\log N(m)}{\sqrt{\log\log N(m)}}\le\gamma\right\}\right|=\Phi(\gamma).$$ The conditions require that large-norm prime ideals rarely divide $m\in S(x)$, that the mean densities $\lambda_\wp$ sum to $\log\log x+o((\log\log x)^{1/2})$, and that the $u$-variable joint divisibility errors $e_{\wp_1\cdots\wp_u}$ are small in a uniform sense. The paper then proves Theorems 1.3 and 1.4 by showing that $h$-free ideals and $h$-full ideals satisfy these conditions, using the new counting estimate $|N_h(x)|=\kappa\gamma_h x^{1/h}+O_h(R_{N_h}(x))$ stated as Theorem 1.2. The contribution is a general condition-checking theorem plus the two worked verifications for sparse ideal families.

Load-bearing premise

The proof relies on, but does not derive, a uniform joint divisibility error bound: when $S(x)$-elements are required to be divisible by several distinct prime ideals at once, the error must stay as small as the product of the per-prime errors, uniformly over all choices of the primes. If that joint error is actually larger, condition (f) fails and the Gaussian conclusion for that $S$ is not established.

Editorial extensions

If this is right

Taking $S=\mathcal{M}$ recovers the Erdős–Kac theorem over number fields, and taking $K=\mathbb{Q}$ recovers the classical Erdős–Kac theorem for integers.
The $h$-free ideals of any number field obey the Gaussian law: the proportion of such ideals with $N(m)\le x$ and normalized $\omega(m)\le\gamma$ tends to $\Phi(\gamma)$.
The $h$-full ideals of any number field obey the same Gaussian law, giving a number-field analogue of the known distribution of powerful integers.
Theorem 1.2 provides the explicit asymptotic $|N_h(x)|=\kappa\gamma_h x^{1/h}+O_h(R_{N_h}(x))$ for $h$-full ideals, which is the counting input behind the $h$-full Gaussian law.
Any newly proposed family whose divisibility frequencies are multiplicative up to the specified uniform error automatically inherits the Gaussian law, so the theorem acts as a reusable template for future sparse families.

Reading between the lines

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

Beyond the paper's proofs, condition (f) is the real obstacle for new families: verifying the uniform joint-error bound is the concrete step that would let the same theorem apply to other structured subsets of ideals.
The paper states that a subset version over arbitrary abelian monoids will appear in a follow-up; translating conditions (a)--(f) to monoid language would make the result fully general, but that translation is not carried out here.
A testable extension would be to families such as ideals whose prime factors lie in prescribed residue classes or with prescribed ramification; the moment-matching method should go through once the joint-error estimate is proved.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The abstract subset theorem is a genuine contribution: it gives a clean set of hypotheses under which an additive function over a subset of ideals in a number field obeys a Gaussian law, and the proof is a transparent adaptation of Liu's monoid framework. The h-free application is plausible and the main counting estimates are standard. But the h-full application has a concrete sign error that invalidates the proof of Theorem 1.4 as written.

The stress-test is right. In Lemma 6.2, the local factor removed from the Euler product should divide the main constant, not multiply it. The generating series for h-full ideals coprime to ℓ is N_h(s)/A_ℓ(s), so the residue at s=1/h is divided by A_ℓ(1/h). The statement and the proof around (32) put the factor in the numerator. The later expression for λ_℘ in (34) is actually the correct one, which suggests a slip rather than a conceptual failure, but as it stands the verification of conditions (d) and (e) for Theorem 1.4 does not go through.

There is also a gap in both applications: the joint divisibility error O(x^τ / ∏ N(℘_i)^τ) is asserted without proof. This uniformity is load-bearing for condition (f). It is probably true, but needs a derivation; the dependence on u is not discussed.

Minor issues: the paper does not compare with the all-purpose theorem in [19] or the authors' own [3], and the constant γ_h in (9) looks miswritten—the displayed Euler product does not match the constant obtained from the zeta factors in the proof of Theorem 1.2.

Net: the framework is worth having, and the h-free part may be correct once the joint error is supplied. The h-full part has a repairable sign error. I would send it to a serious referee, but with a clear request to fix the local factor and the error term. Not ready for acceptance.

Referee Report

2 major / 3 minor

Summary. The paper proves a subset-level Erdős–Kac theorem for ideals of a number field (Theorem 1.1): if a subset S of the ideal monoid satisfies the small-norm negligibility condition (2) and the moment conditions (a)–(f) with a truncation parameter y(x) < x^β, then the normalized counting function (ω(m) − log log N(m))/sqrt(log log N(m)) over m ∈ S, N(m) ≤ x converges in distribution to the standard normal Φ. The proof follows the classical moment method: truncation at y, comparison of ω_y with a sum of independent Bernoulli variables, and the central limit theorem. The paper then applies this criterion to h-free ideals (Theorem 1.3) and h-full ideals (Theorem 1.4), after proving the needed density estimates, including the h-full count in Theorem 1.2.

Significance. If the proofs are completed, the paper provides a useful and flexible criterion: instead of developing a new sieve, it isolates weak moment conditions under which the Erdős–Kac law holds for sparse subsets of ideals, and it demonstrates the criterion on two nontrivial families. The proof of Theorem 1.1 itself is logically coherent and involves no fitted parameters. The h-free application is largely convincing, although one joint-error estimate is asserted without proof. The h-full application, however, contains a concrete factor error in Lemma 6.2 that invalidates the proof of Theorem 1.4 as written. The defect appears repairable, but the current manuscript needs substantive revision before the advertised applications are established.

major comments (2)

[Section 6, Lemma 6.2 and Eqs. (31)–(34)] The local factor in Lemma 6.2 is inverted. With F_ℓ(s) = 1 + N(ℓ)^{-hs}/(1 − N(ℓ)^{-s}), Eq. (31) gives N_{h,ℓ}(s) = N_h(s)/F_ℓ(s). At s = 1/h the density of ℓ-coprime h-full ideals should therefore be κγ_h x^{1/h} / F_ℓ(1/h), i.e. multiplied by (1 − N(ℓ)^{-1/h})/(1 − N(ℓ)^{-1/h} + N(ℓ)^{-1}). The lemma instead states a factor (1 + N(ℓ)^{-1})/(1 − N(ℓ)^{-1/h}), which is larger than 1 for large N(ℓ); for K = Q, h = 2, ℓ = 2 it would predict that odd powerful numbers are about 5.12 times as dense as all powerful numbers, which is impossible. The definition of G_{h,ℓ}(s) in (32) also multiplies by F_ℓ instead of dividing by it, and the final product evaluation in the proof is inconsistent with (31). The value of λ_℘ displayed in (34) is the correct one after the factor is repaired, but it does not follow from the lemma as printed. Since the verification of Conditions (d), (e), and (f) in Theorem 1.4 uses this λ_℘, the h-full application is not proved as written.
[Section 5, after (14); Section 6, after (34)] The verification of Condition (f) in both applications relies on an asserted joint divisibility estimate: for distinct prime ideals ℘_1, …, ℘_u, the count of elements of S(x) divisible by all of them equals the product of the individual main factors times |S(x)| plus an error O(x^τ / ∏ N(℘_i)^τ). This estimate is stated as following from the individual estimates and the Chinese Remainder Theorem, but it is not derived. Lemmas 5.1 and 6.2 handle coprimality to only one prime; after dividing by ℘_1^{k_1}⋯℘_u^{k_u}, one needs a count of h-free or h-full ideals coprime to the full product, with the displayed dependence of the error on each N(℘_i). Because Condition (f) sums these errors over all u-tuples and requires o((log log x)^{-r/2}), this missing uniformity is load-bearing. A multi-prime analogue of Lemmas 5.1 and 6.2, or another explicit bound for the u-tuple error sum, is necessary to complete the proofs of Theorems 1.3 and 1.4.

minor comments (3)

[Section 3, Lemma 3.5] The proof of Lemma 3.5 is deferred to [17, Lemma 7] without a sketch. This is likely standard, but a few lines outlining the moment bound would make the paper more self-contained and would clarify why the hypotheses of Fact 5 and Fact 4 are satisfied.
[Section 6, Eq. (32)] In the definition of G_{h,ℓ}(s), the symbol φ_K^k(s) should be φ_K^h(s); this appears to be a typographical slip but should be corrected.
[Section 5, Eq. (14)] The displayed formula for |S_℘(x)| has an unbalanced parenthesis in the error term O_h(R_{S_h}(x/N(℘)^k))); it should read O_h(R_{S_h}(x/N(℘)^k)).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the subset theorem is conditional, the applications verify its conditions by independent counting arguments, and self-citations only supply a proof template.

full rationale

This paper is not circular. Theorem 1.1 is a conditional statement: if a subset S satisfies conditions (a)-(f), including arithmetic estimates lambda_P and e_P defined by the counting equation (3), then a Gaussian law follows. The proof of the theorem uses [17] as a template for the moment method and CLT framework (Lemmas 3.1-3.5, Section 4), but the conditions are new and the applications verify them by independent counting arguments, not by assuming the desired Gaussian law. The h-free application derives lambda_P from Lemma 5.1, which is based on Landau's ideal-count (6) and the Mobius identity (12), and then checks conditions (a)-(f); the h-full application derives lambda_P from Lemma 6.2 and Theorem 1.2, whose constants come from the Ivic-Shiu identity (18) and zeta-factor arguments. No fitted parameter is renamed as a prediction: the lambda_P are actual asymptotic frequencies, and the joint-divisibility estimates in Sections 5 and 6 are asserted error-uniformity bounds used to verify condition (f), not outputs of the CLT. The paper does contain a real internal algebraic issue in Lemma 6.2, where the final local factor is inconsistent with equations (31)-(32), but that is a correctness defect, not a circular reduction: it does not make the theorem equal to its assumptions. Self-citations to [17] supply the proof framework; they do not carry the central claim by themselves. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on standard analytic number theory inputs (Landau, Mertens, Knopfmacher, Ivić-Shiu) and standard probability facts. The subset-specific hypotheses (2) and (a)-(f) are the paper's assumptions, not external axioms. No fitted parameters or invented entities appear.

assumptions (5)

standard math Landau's prime ideal theorem: π_K(x) = Li(x) + O(x exp(-c_K sqrt(log x)))
Used in Section 3 to derive Mertens-type estimates (Lemma 3.6 Parts 1, 3, 4).
standard math Landau's ideal counting theorem: I_K(x) = κ x + O(x^{1-2/(n_K+1)})
Used in Lemma 5.1, Lemma 6.1, and the definition of κ.
domain assumption Knopfmacher's Axiom A and Proposition 5.5 give the h-free count |S_h(x)| = κ/ζ_K(h) x + O_h(R_{S_h}(x))
Invoked at the start of Section 5; depends on the Dedekind zeta function having a simple pole with residue κ.
standard math Ivić-Shiu identity (18): (1 + v^h/(1-v)) ∏_{i=0}^{h-1} (1-v^{h+i}) = 1 - v^{2h+2} + Σ α_{r,h} v^r
Used to factor the h-full Dirichlet series in Section 6.
standard math Standard probability results (Facts 1-5): convergence in probability, moment convergence implies distribution convergence, and a special CLT for uniformly bounded independent random variables
Section 2 and used throughout Section 4.

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Pith. "Pith review of A subset generalization of the Erd\H{o}s-Kac theorem over number fields with applications." pith.science (2026). https://pith.science/paper/TRRENYMO

@misc{pith2026250603215,
  author       = {Pith},
  title        = {Pith review of: A subset generalization of the Erd\Hos-Kac theorem over number fields with applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRRENYMO}},
  note         = {Machine review of arXiv:2506.03215}
}

abstract

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erd\H{o}s and Kac established that $\omega(n)$ obeys the Gaussian distribution over natural numbers. In 2004, the third author generalized their theorem to all abelian monoids. In this work, we extend the work of the third author to any subset of the set of ideals of a number field satisfying some additional conditions. Finally, we apply this theorem to prove the Erd\H{o}s-Kac theorem over $h$-free and over $h$-full ideals of the number field.

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Reference graph

Works this paper leans on

19 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [19]

M. R. Murty, V. K. Murty, and S. Pujahari. An all-purpose Erdős-Kac theorem. Math. Z. , 305(3):Paper No. 45, 18, 2023. 33

work page 2023

[2] [3]

S. Das, W. Kuo, and Y.-R. Liu. Distribution of ω(n) over h-free and h-full numbers. International Journal of Number Theory , pages 1–23, 2025

work page 2025

[3] [1]

Alladi and P

K. Alladi and P. Erdős. On an additive arithmetic functio n. Paciﬁc J. Math. , 71(2):275–294, 1977. 31

work page 1977

[4] [2]

Billingsley

P. Billingsley. On the central limit theorem for the prim e divisor functions. Amer. Math. Monthly , 76:132–139, 1969

work page 1969

[5] [4]

Erdős and M

P. Erdős and M. Kac. The Gaussian law of errors in the theor y of additive number theoretic functions. Amer. J. Math. , 62:738–742, 1940

1940

[6] [5]

J. A. Gómez and M. Lalín. Prime factors with given multipl icity in h-free and h-full polynomials over function ﬁelds. Pre-print, 2023

2023

[7] [6]

Granville and K

A. Granville and K. Soundararajan. Sieving and the Erdős -Kac theorem. In Equidistribution in number theory, an introduction , volume 237 of NATO Sci. Ser. II Math. Phys. Chem. , pages 15–27. Springer, Dordrecht, 2007

work page 2007

[8] [7]

Halberstam

H. Halberstam. On the distribution of additive number-t heoretic functions. J. London Math. Soc. , 30:43–53, 1955

work page 1955