REVIEW 4 major objections 4 minor 13 references
Generalizations of Erd\H{o}s-Kac theorem with applications
T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper establishes an Erdős-Kac theorem for arbitrary subsets of abelian monoids satisfying density and independence conditions, and applies it to h-free and h-full elements, yielding Gaussian laws for ω, Ω, log d, and related…
desk verdict Solid monoid-level Erdős–Kac framework undermined by an incorrect h-full density lemma quoted from an unpublished preprint. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is a moment-matching argument. For a truncation level $y$, the paper forms the random variable $S_y = \sum_{N(\ell)\le y} X_\ell$, where the $X_\ell$ are independent Bernoulli variables with success probabilities $\lambda_\ell$. Condition (d) forces $E(S_y) = \log\log x + o((\log\log x)^{1/2})$ and condition (e) forces $\mathrm{Var}(S_y) = \log\log x + o((\log\log x)^{1/2})$. Lemma 3.4 shows that the normalized moments of the deterministic truncated counter $\omega_y(f(m))$ match those of $S_y$ exactly in the limit, and Lemma 3.5 bounds them so that a standard central limit theorem (Fact 5) applies. The error terms in conditions (b), (c), and (f) are exactly what is needed to kill the difference between $\omega_y(f(m))$ and $\omega(f(m))$ and to kill the higher-order correlations, so the Gaussian law transfers from the independent model $S_y$ to the real counting function.
What would settle it
One concrete test: for the h-full integers with h=2, compare the actual proportion of n ≤ x with (ω(n) − log log n)/√(log log n) ≤ γ against Φ(γ) for fixed γ; Theorem 1.4 predicts the difference tends to zero, so a persistent discrepancy would show either the theorem or the counting estimate (6) is wrong. A sharper check is to test condition (f) directly: for all distinct pairs of primes p,q ≤ $x^{{1/2}}$/log log x, compare |{n ≤ x : n h-full, p|n, q|n}| / |N_h(x)| with 1/(N(p)N(q)); the proof requires the summed deviations over all such pairs to be o((log log x)^{−r/2}) for every fixed r.
Extended reading notes
Core claim
The central discovery is Theorem 1.1: if a subset $S$ of a free abelian monoid $M$ satisfies condition (2) and conditions (a)--(f) on the densities $\lambda_\ell$ and errors $e_\ell$ of the events $\{n_\ell(f(m)) \ge 1\}$, then for every real $\gamma$ the proportion of $m \in S(x)$ with $(\omega(f(m)) - \log\log N(m))/\sqrt{\log\log N(m)} \le \gamma$ tends to $\Phi(\gamma)$. The point is that the normalization uses $\log\log N(m)$, the same as in the classical theorem, even when $S$ is sparse, as long as the truncated small-prime events behave almost independently. Theorem 1.2 is a less abstract but easily checkable variant: if $|S(x)| = C_\beta x^\beta + O(x^{\xi\beta})$ and the count of elements whose $f$-image contains $p$ is $(C_\beta/N(p))x^\beta$ with a secondary term uniformly bounded in $p$, then the Gaussian limit follows. The remaining theorems and corollaries verify these hypotheses for h-free and h-full elements and for the maps $f_k$ producing the functions $\omega_k$, $\Omega$, $\log d$, and $\omega_T$.
Load-bearing premise
The load-bearing premise is that for the chosen truncation level y, the events 'a randomly picked element of S maps under f to something divisible by the small prime ℓ' are almost independent across ℓ, with the cumulative correlation error in conditions (e) and (f) tending to zero faster than any negative power of log log x.
Editorial extensions
If this is right
- The classical Erdős-Kac theorem and its abelian-monoid version follow as the special case $S=M$, $f=\mathrm{identity}$.
- The normalized number of distinct prime factors $\omega(m)$ has a Gaussian limit over h-free and h-full elements in any monoid satisfying the norm-counting axiom.
- On k-full elements, the functions $\omega_k$ (prime factors of exact multiplicity k), $\Omega(m)/k$, and $\log d(m)/\log(k+1)$ all obey the same Gaussian limit, with the same mean $\log\log N(m)$.
- For any coefficient sequence $A$ with $a_k \neq 0$ and fast-decaying tail, the normalized weighted function $(1/a_k)\omega_A(m)$ also obeys the Gaussian law on k-full elements.
- The same conclusions hold for h-free and k-full ideals in number fields, effective divisors in global function fields, and effective 0-cycles on geometrically irreducible projective varieties.
Reading between the lines
- The specific corollaries for h-free and h-full elements lean on the h-full counting estimate (6) and on Lemmas 3.7 and 3.8, which the paper quotes from a companion submitted manuscript; if any of those inputs turn out to need a stronger error term, the corresponding applications would weaken, even though Theorem 1.1 itself would remain valid.
- A testable extension would check the local-density condition (12) for other sparse multiplicative sets, such as y-smooth elements or shifted primes, where the same moment method would yield a Gaussian law if the correlation errors are controlled.
- Because the mean remains $\log\log N(m)$ for h-full elements, the Gaussian law is driven by many small primes rather than by the density of the set; this suggests the phenomenon is insensitive to the exponent $\beta$ in $|S(x)| \sim C_\beta x^\beta$.
- For h-free elements Theorem 1.8 assumes $a_1 \neq 0$; the case where the first non-zero coefficient sits at some $k>1$ is not treated and would need its own tail estimate.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents a general Erdős-Kac theorem for counting functions ω(f(m)) over subsets S of free abelian monoids satisfying a collection of explicit conditions (⋆), (2), and (a)–(f). The main theorem (Theorem 1.1) is proved by the standard moment method via truncated sums and comparison with independent Bernoulli random variables, and it yields a densitometric version (Theorem 1.2). The authors then derive Erdős-Kac theorems for ω over h-free and h-full elements (Theorems 1.3–1.4), for the generalized functions ω_k, ω_A, Ω, log d, and ω_T over k-full elements (Theorems 1.5–1.8 and Corollaries 1.1–1.3), and close with applications to ideals in number fields, effective divisors in global function fields, and 0-cycles on projective varieties.
Significance. If fully correct, the framework would be a useful unification of Erdős-Kac-type results in abstract analytic number theory, extending the third author's earlier monoid theorem to arbitrary subsets with controlled density and to a wide family of arithmetic functions via a single sequence A. The conditions in Theorem 1.1 are explicit and checkable, and the extensive list of applications shows the intended reach of the result. However, the present version contains technical errors in the h-full applications and in the statements of the complex-coefficient theorems, so the significance can only be assessed after these are repaired.
major comments (4)
- [Lemma 3.8] Lemma 3.8 is internally inconsistent as stated. For r=1, the displayed factor (1+N(ℓ)^{-1})/(1-N(ℓ)^{-1/h}) is greater than 1 for every prime ℓ (e.g., about 5.12 for N(ℓ)=2, h=2), so the set of h-full elements avoiding ℓ would have a larger asymptotic density than the full h-full set counted in (6). The decomposition m=ℓ^a m' used in the proof of Theorem 1.4 requires the factor to be (1-N(ℓ)^{-1/h})/(1-N(ℓ)^{-1/h}+N(ℓ)^{-1}), which is less than 1. Consequently the displayed equality in the proof of Theorem 1.4, identifying Σ_{k=h}^∞ N(ℓ)^{-k/h} κγ_h(1+N(ℓ)^{-1})/(1-N(ℓ)^{-1/h}) with κγ_h/(N(ℓ)(1-N(ℓ)^{-1/h}+N(ℓ)^{-1})), is algebraically false with the printed lemma. The same inconsistency appears in the proof of Theorem 1.6, where the first line uses the printed factor but the final C'_{p,β} is derived from the corrected factor. Theorems 1.4, 1.6, 1.7 and the h-full corollaries are therefore not supported as written; the authors must correct Lemma 3.8 and the subsequent algebra, and re-examine the sign of C'_{p,β} in the proofs of Theorems 1.4 and 1.6.
- [Theorems 1.7 and 1.8] Theorems 1.7 and 1.8 are stated for a sequence A=(a_i) of complex numbers, but the conclusions involve the inequality (1/a_k)ω_A(m) ≤ a (and similarly with 1/a_1), which is not meaningful when the a_i are complex. The proof of Theorem 1.7 uses real-valued inequalities: after defining G_g in (17), the argument relies on implications such as G_{ω_k}(m)-ε ≤ G_{(1/a_k)ω_A}(m) ≤ G_{ω_k}(m)+ε and the sandwiching in (19)–(22). These steps require (1/a_k)ω_A(m) to be real. The manuscript should either restrict A to real sequences with a_k ≠ 0 (as in the concrete applications in Corollaries 1.1–1.3 and Section 7) or provide a genuinely complex version with a different target statistic.
- [Proof of Theorem 1.2] In the proof of Theorem 1.2, after setting y=x^β/log log x, the authors state that 'the set in Condition (a) is empty' and hence condition (a) holds trivially. For a general map f:S→M this is false: if f is the identity and m is a prime element with N(m)=x, then n_ℓ(f(m))≥1 for ℓ=m with N(ℓ)=x>x^β, so the set is nonempty. Condition (a) does hold in all applications because for those f (identity or f_k(m)=m_k) one has N(f(m))≤N(m)≤x, which yields the needed O_β(1) bound on the number of primes of norm >x^β. The statement of Theorem 1.2 should include a norm-growth hypothesis such as N(f(m))≪N(m)^C for m∈S(x), and the proof should verify (a) through that property rather than by asserting emptiness.
- [Equations (3), (6), Lemmas 3.7, 3.8] The h-free and h-full counting estimates used in the applications are quoted from the authors' submitted preprint [1], which is not publicly available. This dependence is already a verification concern, and it is compounded by the fact that Lemma 3.8, as quoted, is demonstrably false (see the first major comment). The h-free estimate (3) is standard, so the main risk is concentrated in (6) and Lemma 3.8. The authors should provide complete proofs of the quoted auxiliary estimates in this paper, or at least a detailed appendix, so that the support for Theorems 1.4–1.8 and the h-full corollaries can be checked.
minor comments (4)
- [Remark 7.1] In Remark 7.1, 'Erdos' should be 'Erdős' to match the spelling elsewhere in the paper.
- [Lemma 3.8] The product formula in Lemma 3.8 is ambiguous: it should be made clear whether (κγ_h)^r multiplies the product of the ratios or κγ_h multiplies the product of the ratios; the proof only uses r=1, but the statement for general r should be unambiguous.
- [Section 1] The phrase 'Let S be a subset of infinitely many elements in M' should be reworded to 'an infinite subset of M'.
- [Theorem 1.8] The proof of Theorem 1.8 is omitted with the note that it is analogous to Theorem 1.7. Since the proof of Theorem 1.7 requires correction due to the real-coefficient issue, the authors should include at least a sketch of the proof of Theorem 1.8 or state explicitly which steps differ.
Circularity Check
No circular reduction found; the heavy reliance on the authors' own submitted counting lemmas is a support concern rather than a circularity.
full rationale
The central theorem (Theorem 1.1) is proved directly and self-containedly: the quantities λℓ and eℓ are defined from the subset S, and conditions (a)–(f) are used to show, through the helper lemmas of Section 3, that the moments of the normalized ω(f(m)) converge to the Gaussian moments. The Gaussian law is the output of a central-limit argument, not an assumed input. The applications to h-free and h-full elements use counting asymptotics (3), (6) and Lemmas 3.7–3.8 quoted from the authors' submitted preprint [1]; these are auxiliary density estimates and do not themselves assert the Erdős-Kac conclusion, so no equation is used as both hypothesis and conclusion. The self-citations are therefore not circular in the technical sense. A separate correctness issue is that Lemma 3.8, as printed, is internally inconsistent: for a fixed prime ℓ, the factor (1+N(ℓ)−1)/(1−N(ℓ)−1/h) exceeds 1, so the set of h-full elements avoiding ℓ would have larger asymptotic density than the full h-full set, undermining Theorems 1.4, 1.6, and 1.7. This is a support defect in the quoted counting input, not a circular reduction, and it does not affect the self-contained proof of Theorem 1.1 itself.
Assumptions & free parameters
assumptions (6)
- domain assumption Axiom A (⋆): M(x) = ∑_{m∈M, N(m)≤x} 1 = κx + O(x^θ) for some κ>0 and 0≤θ<1.
- domain assumption Counting asymptotic for h-free elements: |S_h(x)| = κ/ζ_M(h) x + O_h(R_{S_h}(x)).
- domain assumption Counting asymptotic for h-full elements: |N_h(x)| = κγ_h x^{1/h} + O_h(R_{N_h}(x)).
- domain assumption Mertens-type theorem: ∑_{N(p)≤x} 1/N(p) = log log x + A + O(1/log x).
- domain assumption Local density lemmas for h-free and h-full elements avoiding fixed primes.
- domain assumption Zeta function ζ_M(s) converges absolutely for Re(s)>1 and has a simple pole at s=1 (implicit in Knopfmacher's framework).
Cite this review
Pith. "Pith review of Generalizations of Erd\H{o}s-Kac theorem with applications." pith.science (2026). https://pith.science/paper/C4QERRUW
@misc{pith2026250602432,
author = {Pith},
title = {Pith review of: Generalizations of Erd\Hos-Kac theorem with applications},
year = {2026},
howpublished = {\url{https://pith.science/paper/C4QERRUW}},
note = {Machine review of arXiv:2506.02432}
}
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, and in 2004, the third author generalized their theorem to all abelian monoids. In this paper, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of $h$-free and $h$-full elements. We study generalizations of several arithmetic functions, such as the prime counting omega functions and the divisor function in a unified framework. Finally, we apply our results to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of our approach.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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