REVIEW 2 major objections 5 minor 19 references
On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid
T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read For any countably generated free abelian monoid whose norm-counting function satisfies a standard power-saving estimate, the number of distinct prime generators of h-free and h-full elements has explicit first and second moments, and…
desk verdict The h-free part is a solid generalization, but the h-full moment proof is broken by a wrong factor in Lemma 4.2. 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 the generalized zeta function $\zeta_M(s)=\prod_{p\in P}(1-N(p)^{-s})^{-1}$, together with two structural decompositions. For $h$-free elements, the generalized Möbius function converts the indicator of $h$-freeness into a divisor sum, and Lemma 3.1 counts $h$-free elements avoiding any fixed finite set of primes; feeding in the estimate $\sum_{N(p)\le x}1/N(p)=\log\log x+A+O(1/\log x)$ produces the double-log main terms and the constants $C_1,C_2$. For $h$-full elements, every element is written as $m=h a_0+(h+1)a_1+\cdots+(2h-1)a_{h-1}$, and the generating series $N_h(s)$ factors as $G_h(s)L_h(s)$, where $L_h(s)=\zeta_M(hs)\zeta_M((h+1)s)\cdots\zeta_M((2h-1)s)$ and $G_h(s)$ converges absolutely for $\Re(s)>1/(2h+2)$; evaluating this factorization at $s=1/h$ yields the Euler product $\gamma_h$ and the constants $D_1,D_2$. Condition $(\star)$ is used throughout to justify the zeta-function analysis and the error terms.
What would settle it
Take $M$ to be the free abelian monoid on the ordinary primes with $N(p)=p$, and take $h=2$, so that $h$-free elements are the squarefree integers and $\zeta_M(2)=\pi^2/6$. Theorem 1.2 predicts $\sum_{n\le x,\ n\ \mathrm{squarefree}}\omega(n)=\frac{6}{\pi^2}x\log\log x+\frac{6}{\pi^2}C_1 x+O(x/\log x)$, with $C_1=A-\sum_p\frac{p-1}{p(p^2-1)}$. Directly evaluating the squarefree sum for large $x$ should show the residual after subtracting $\frac{6}{\pi^2}x\log\log x$ tracking the linear term $\frac{6}{\pi^2}C_1 x$; a persistent mismatch in that coefficient would disprove the theorem.
Extended reading notes
Core claim
The central claim is Theorem 1.2 and Theorem 1.3. If $P$, $M$, and $X$ satisfy Condition $(\star)$, namely $I(x)=\sum_{N(m)\le x}1=\kappa x+O(x^\theta)$ with $\kappa>0$ and $0\le\theta<1$, then $\sum_{m\in S_h(x)}\omega(m)=\frac{\kappa}{\zeta_M(h)}x\log\log x+\frac{\kappa C_1}{\zeta_M(h)}x+O_h(x/\log x)$, with a corresponding explicit second moment involving $C_2$, and $\sum_{m\in N_h(x)}\omega(m)=\kappa\gamma_h x^{1/h}\log\log x+\kappa\gamma_h D_1 x^{1/h}+O_h(x^{1/h}/\log x)$, with the second moment involving $D_2$. The constants $C_1,C_2,D_1,D_2$ are explicit Euler-product and reciprocal-prime sums depending only on the prime set and $h$, and $\gamma_h$ is the convergent Euler product of (4). Corollaries 1.1 and 1.2 turn these moments into the normal-order statement: for every $\varepsilon>0$, all but $o(|S_h(x)|)$ $h$-free elements and all but $o(|N_h(x)|)$ $h$-full elements have $(1-\varepsilon)\log\log N(m)\le\omega(m)\le(1+\varepsilon)\log\log N(m)$. Section 5 verifies Condition $(\star)$ in three concrete settings and writes out the resulting formulas for number fields, function fields, and projective varieties.
Load-bearing premise
The load-bearing premise is Condition $(\star)$: the count of monoid elements with norm at most $x$ must be $\kappa x+O(x^\theta)$ for some $\kappa>0$ and $0\le\theta<1$; if this estimate fails, or its error is not a pure power strictly below $x$, the stated leading constants and the normal-order conclusion need not survive.
Editorial extensions
If this is right
- For both $h$-free and $h$-full subsets, $\omega(m)$ has normal order $\log\log N(m)$: for any $\varepsilon>0$, the fraction of elements in $S_h(x)$ or $N_h(x)$ whose $\omega(m)$ lies outside $(1\pm\varepsilon)\log\log N(m)$ tends to 0.
- The second-moment formulas imply the variance of $\omega$ over each subset grows like $\log\log x$, matching the classical integer picture and making a Gaussian limit plausible; the paper states that a Gaussian distribution result will appear in a follow-up.
- The same theorems give explicit moment formulas for ideals in number fields, effective divisors in global function fields, and effective 0-cycles on geometrically irreducible projective varieties; the number-field case with $K=\mathbb{Q}$ recovers the earlier integer results.
- An analogous treatment of $\Omega(m)$, the count with multiplicity, yields first and second moments over $h$-free and $h$-full elements and normal order $\log\log N(m)$ over $M$ and $S_h$, with normal order $h\log\log N(m)$ over $N_h$.
Reading between the lines
- An extension the paper leaves implicit: because the proofs use Condition $(\star)$ only through the zeta-function and reciprocal-prime estimates, the same moment formulas should hold for any abstract number system satisfying the same power-saving counting axiom, including systems whose prime norms are not powers of a single base.
- The explicit constants $C_2$ and $D_2$ determine the variance of $\omega$ over each subset up to a bounded error; if the announced Gaussian distribution result is proved, these constants fix the variance's constant term and give a way to test the rate of convergence in number-field and function-field examples.
- The $h$-full result says that although $h$-full elements have density zero, their distinct-prime counts spread exactly like the whole monoid; this suggests the decomposition $m=h a_0+(h+1)a_1+\cdots$ can be used as a randomizing model for other arithmetic functions over sparse subsets.
- In the function-field and variety cases the constants become explicit functions of $q$ and the zeta function of the underlying object; evaluating $\gamma_h,C_1,C_2,D_1,D_2$ for small genus or small dimension would yield concrete asymptotic predictions checkable against divisor-counting data.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper develops a general framework for counting h-free and h-full elements in countably generated free abelian monoids equipped with a norm map and a counting function satisfying the asymptotic condition (⋆). The main theorems give asymptotic formulas for the first and second moments of the number of distinct prime generators over h-free and h-full elements, with explicit constants C1, C2, D1, D2 and specified error terms, and they derive the normal order log log N(m) on both subsets. Section 5 applies the results to ideals in number fields, effective divisors in global function fields, and effective 0-cycles on geometrically irreducible projective varieties over finite fields.
Significance. If the moment formulas are correct, the paper provides a useful unified treatment of ω over h-free and h-full elements in a broad class of arithmetical monoids. The constants are defined by convergent Euler products or sums rather than fitted to the claimed asymptotics, and the applications verify the key condition (⋆) by reference to concrete counting theorems. The normal-order corollaries and the transfer to number fields, function fields, and varieties are natural and would be of interest. However, the manuscript contains a load-bearing algebraic error in Lemma 4.2 that invalidates the proof of Theorem 1.3 as printed; the intended argument appears repairable by a reciprocal correction, but the current text does not establish the stated D1 and D2.
major comments (2)
- [Section 4, Lemma 4.2] The stated leading constant has the exclusion factor in the numerator, but the restricted generating function is N_{h,ell}(s) = N_h(s) / (1 + N(ell)^{-hs}/(1 - N(ell)^{-s})) by (27). Therefore removing the prime ell divides by the factor (1 + N(ell)^{-1}/(1 - N(ell)^{-1/h})) at s = 1/h, not multiplies by it. This is also visible in the final displayed product of the proof: prod_{p neq ell} A_p times prod_p B_p equals gamma_h / A_ell, not gamma_h A_ell, where A_p = 1 + N(p)^{-1}/(1 - N(p)^{-1/h}). Lemma 4.2 and its proof must be corrected by replacing the factor with its reciprocal.
- [Section 4, proof of Theorem 1.3, Eqs. (45)-(46)] With the printed multiplicative factor A = 1 + N(p)^{-1}/(1 - N(p)^{-1/h}), the inner sum in (45) is sum_{k=h}^{floor(...)} A N(p)^{-k/h}, whose leading term is N(p)^{-1}(1 - N(p)^{-1/h} + N(p)^{-1})/(1 - N(p)^{-1/h})^2, not N(p)^{-1}/(1 - N(p)^{-1/h} + N(p)^{-1}) as claimed in (46). The displayed simplification is therefore false, and the derivation of D1 collapses; the same erroneous factor propagates into the pair sum and the derivation of D2 in (52)-(56). Replacing the factor in Lemma 4.2 by its reciprocal makes the identity in (46) correct, so the theorem is plausibly repairable, but Theorem 1.3 is not proved by the text as written.
minor comments (5)
- [Section 4.1, Eqs. (5) and (36)] The case conditions involving h/(h+i) are typeset ambiguously; please use h/(h+i) with parentheses consistently in both remainder definitions.
- [Section 4.1, Lemma 4.1] Lemma 4.1 is attributed to [3, Lemma 6.1] even though a full proof is supplied in the text; please clarify whether this is a new proof or a reproduction of the cited lemma.
- [Section 2, proof of Lemma 2.5] The intermediate expression A(log log x/2 + A) appears to be a typo; it should presumably read A(log log(x/2) + A).
- [Section 3, Eq. (20)] The denominator zeta(h) in (20) should be zeta_M(h) to match the notation used throughout the paper.
- [Section 1, Corollary 1.2] The proof of Corollary 1.2 is only a sketch referencing [4, Proof of Theorem 1.3]; once Theorem 1.3 is repaired, please spell out the variance step or provide a precise reduction to the moment estimates.
Circularity Check
No load-bearing circularity: constants are defined by convergent prime sums and independent Axiom-A estimates, not fitted to the target moments.
full rationale
The derivation chain is not circular. Under Condition (⋆), the main constants γ_h, C1, C2, D1, D2 are defined in advance by convergent Euler products, limits, or prime sums ((4), (6)–(12)); none is chosen to match the right-hand sides of Theorems 1.2–1.3. The h-free moment proof expands the counting function via Möbius inversion (Lemma 3.1) and reduces to Mertens-type prime estimates; the h-full proof factors the generating series using the external identity from Ivić–Shiu [8, (1.5)] and the condition-(⋆) count T_h(x) proved in Lemma 4.1. The self-citations to [16] and [17] supply a classification of X and the constants A and B under the same Axiom A hypothesis; these are parameter-free published results whose assumptions do not include the target moments, so by the independence rule they do not make the claim circular. The integer case [4] serves as an external benchmark that the monoid results generalize, rather than as the source of the monoid constants. The applications are not fitted: they verify Condition (⋆) in each setting via Landau, Rosen, and [17, Lemma 7]. The apparent reciprocal-factor mismatch in Lemma 4.2 is a technical correctness issue in a non-circular step, not an equivalence of output to input, and it does not affect the circularity score.
Assumptions & free parameters
assumptions (4)
- domain assumption Condition (⋆): I(x) = Σ_{m∈M, N(m)≤x} 1 = κx + O(x^θ) with κ > 0 and 0 ≤ θ < 1.
- standard math X is either Q or {q^z : z ∈ Z} for some q > 1.
- standard math Absolute convergence and Euler product for ζ_M(s) for Re(s) > 1, with nonzero values.
- domain assumption Application-specific counting estimates: Landau's prime ideal theorem for number fields; Rosen [19] for effective divisors; [17, Lemma 7] for 0-cycles on projective varieties.
Cite this review
Pith. "Pith review of On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid." pith.science (2026). https://pith.science/paper/FLVC43LU
@misc{pith2026250601030,
author = {Pith},
title = {Pith review of: On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid},
year = {2026},
howpublished = {\url{https://pith.science/paper/FLVC43LU}},
note = {Machine review of arXiv:2506.01030}
}
read the original abstract
This work introduces the first in-depth study of h-free and h-full elements in abelian monoids, providing a unified approach for understanding their role in various mathematical structures. Let m be an element of an abelian monoid, with {\omega}(m) denoting the number of distinct prime elements generating m. We study the moments of {\omega}(m) over subsets of h-free and h-full elements, establishing the normal order of {\omega}(m) within these subsets. Our findings are then applied to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of this approach.
Reference graph
Works this paper leans on
-
[4]
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
2025
-
[1]
K. Alladi and P. Erdős. On an additive arithmetic functio n. Pacific J. Math. , 71(2):275–294, 1977
work page 1977
-
[2]
S. Das, W. Kuo, and Y.-R. Liu. On the number of prime factor s with a given multiplicity over h-free and h-full numbers. J. Number Theory 267, 176–201, 2025
work page 2025
-
[3]
S. Das, W. Kuo, and Y.-R. Liu. A subset generalization of t he Erdős-Kac theorem over number fields with applications. Submitted, 2024
2024
-
[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
-
[6]
G. H. Hardy and S. Ramanujan. The normal number of prime fa ctors of a number n [Quart. J. Math. 48 (1917), 76–92]. In Collected papers of Srinivasa Ramanujan, pages 262–275. AMS Chelsea Publ., Providence, RI, 2000
work page 1917
-
[7]
G. H. Hardy and E. M. Wright. An introduction to the theory of numbers . Oxford University Press, Oxford, sixth edition, 2008. Revised by D . R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles
work page 2008
-
[8]
A. Ivić and P. Shiu. The distribution of powerful integer s. Illinois J. Math. , 26(4):576–590, 1982. 37
work page 1982
Show all 19 references
-
[9]
Jakimczuk and M
R. Jakimczuk and M. Lalín. The number of prime factors on a verage in certain integer sequences. J. Integer Seq. , 25(2):Art. 22.2.3, 15, 2022
2022
-
[10]
Jakimczuk and M
R. Jakimczuk and M. Lalín. Sums of ω(n) and Ω( n) over the k-free parts and k-full parts of some particular sequences. Integers, 22:Paper No. A113, 22, 2022
2022
-
[11]
Knopfmacher
J. Knopfmacher. Analytic arithmetic of algebraic function fields , volume 50 of Lecture Notes in Pure and Applied Mathematics . Marcel Dekker, Inc., New York, 1979
1979
-
[12]
Knopfmacher
J. Knopfmacher. Abstract analytic number theory . Dover Books on Advanced Mathematics. Dover Publications, Inc., New York, second ed ition, 1990
1990
-
[13]
Lalín and Z
M. Lalín and Z. Zhang. The number of prime factors in h-free and h-full poly- nomials over function fields. Publ. Math. Debrecen , 104(3-4):377–421, 2024
2024
-
[14]
E. Landau. Neuer Beweis des Primzahlsatzes und Beweis d es Primidealsatzes. Math. Ann., 56(4):645–670, 1903
1903
-
[15]
E. Landau. Einführung in die elementare und analytische Theorie der al gebrais- chen Zahlen und der Ideale . Chelsea Publishing Co., New York, 1949
1949
-
[16]
Y.-R. Liu. A generalization of the Erdős-Kac theorem an d its applications. Canad. Math. Bull. , 47(4):589–606, 2004
2004
-
[17]
Y.-R. Liu. A generalization of the Turán theorem and its applications. Canad. Math. Bull. , 47(4):573–588, 2004
2004
-
[18]
Lorenzini
D. Lorenzini. An invitation to arithmetic geometry , volume 9 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 1996
1996
-
[19]
M. Rosen. Number theory in function fields , volume 210 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. 38
2002
Reviewed August 7, 2026 · model on record in the stance chip above.
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