REVIEW 2 major objections 4 minor 8 references
Rigidity of Averages over the Two Largest Prime Factors
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read If the average of $f(P_1(n))$ converges, then the average of $f(P_2(n))$ converges to the same limit.
desk verdict A clean and likely correct resolution of the Alladi–Johnson rigidity question, held back only by a fixable truncation typo in the convolution propositions. 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 pair of Dickman kernels \(K_1,K_2\) together with the vague convergence of the translated prime measure \(\Lambda_{f,t}\). \(K_1(u)=\rho(e^u-1)\) for \(u\ge 0\) and \(0\) otherwise, where \(\rho\) is the Dickman function satisfying \(u\rho'(u)+\rho(u-1)=0\); \(K_2(u)=\$int_1^{{e^u-1}}$\rho(e^u-1-v)\,\frac{dv}{v}\) for \(u\ge\log 2\). The load-bearing facts are the convolution identities \(A_j(X_t;f)=\int K_j(t-s)\,d\Lambda_f(s)+o(1)\), the normalization \(\int K_j=1\), the tail estimates \(\sum_j \sup_{j\le u<j+1}K_j(u)<\infty\), and the Fourier nonvanishing \(\widehat{K_1}(\xi)\ne 0\) for every real \(\xi\). The nonvanishing makes the translates of \(K_1\) dense in \($L^{1}$(\mathbb{R})\), so Wiener's theorem converts the relation \(K_1\ast h\equiv\kappa\) into \(h\equiv\kappa\) almost everywhere.
What would settle it
Numerically compute \(\widehat{K_1}(\xi)=\int_0^\infty \rho(e^u-1)$e^{{-i\xi u}}$\,du\) for real \(\xi\) and look for a zero; any real zero would break the Wiener step. Equivalently, any bounded \(f\) with \(A_1(x;f)\to\kappa\) but \(A_2(x;f)\) not converging to \(\kappa\) would refute Theorem 1.1.
Extended reading notes
Core claim
The paper's central discovery is a rigidity statement at the level of the measure \(\Lambda_{f,t} = \sum_p \frac{f(p)}{p}\delta_{\log\log p - t}\): if \(A_1(X_t;f)\to\kappa\) for \(X_t=\exp(e^t)\), then \(\Lambda_{f,t}\to\kappa\, du\) vaguely as \(t\to\infty\). This is shown by writing \(A_1\) and \(A_2\) as convolutions with the Dickman kernels \(K_1(u)=\rho(e^u-1)\) and \(K_2(u)=\$int_1^{{e^u-1}}$\rho(e^u-1-v)\frac{dv}{v}\), up to \(o(1)\). The convolution identity for \(A_1\) and the nonvanishing of \(\widehat{K_1}\) on the real line imply, by Wiener's Tauberian theorem, that every subsequential limit of \(\Lambda_{f,t}\) is the constant \(\kappa\) times Lebesgue measure. The convolution identity for \(A_2\), together with \(\int K_2 = 1\), then gives \(A_2(X_t;f)\to\kappa\).
Load-bearing premise
The argument depends on an externally cited uniform estimate \(\Psi_2(x,y)\ll x\log y/\log x\) for integers whose second-largest prime factor is small, and on reading the truncation parameter \(y\) as \($x^{{e^{-U}}$}\) rather than the printed \(x $e^{{-U}}$\) in the tail estimates of Propositions 4.1 and 6.1.
Editorial extensions
If this is right
- If \(\frac{1}{x}\sum_{n\le x}f(P_1(n))\) converges, then \(\frac{1}{x}\sum_{n\le x}f(P_2(n))\) converges to the same value, with the convention \(f(P_2(n))=0\) for prime powers.
- No bounded function on the primes can produce two different limiting constants for the first- and second-largest prime factor averages.
- Under the same hypothesis, for every \(0<\alpha<\beta\), the weighted prime sum \(\sum_{x^\alpha<p\le x^\beta} f(p)/p\) converges to \(\kappa\log(\beta/\alpha)\).
- The stronger local conclusion \(\Lambda_{f,t}\to\kappa\,du\) means the weighted distribution of primes itself becomes uniformly distributed on the \(\log\log\)-scale once the first average converges.
Reading between the lines
- The same Wiener–kernel strategy may extend to the third-largest distinct prime factor, and likely to any fixed rank, if the corresponding kernel built from the Dickman function has a real-zero-free Fourier transform.
- The theorem suggests that any bounded statistic of the prime factor configuration that depends continuously on the \(\log\log\)-scale positions of the large primes will inherit convergence from the first-largest factor average.
- A quantitative strengthening could come from proving a zero-free strip for \(\widehat{K_1}\); that would turn Wiener's qualitative theorem into explicit error terms in the convergence of \(A_2\).
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for any bounded function f on the primes, convergence of the average A1(x;f) of f(P1(n)) over n≤x implies convergence of A2(x;f), the corresponding average over the second-largest prime factor P2(n), to the same limit. The proof translates both averages into convolutions with explicit Dickman-type kernels on the log-log scale, shows that the weighted prime measure Λ_{f,t} is vaguely precompact, and uses Wiener's Tauberian theorem after proving that the Fourier transform of the first kernel has no real zeros. Vague convergence of Λ_{f,t} to κ du is then passed through the second convolution formula to obtain A2(x;f)→κ. An appendix supplies an alternative smoothing proof of the Tauberian step.
Significance. If correct, Theorem 1.1 settles the Alladi–Johnson question in the negative: no bounded f can give different limiting averages for the largest and second-largest prime factors. The method is attractive and likely transferable: the rigidity is encoded in explicit kernels, and the key technical facts—the no-real-zeros argument for the Dickman-kernel Fourier transform and the kernel-convergence lemma—are clean and self-contained. The paper contains no fitted parameters and the central result does not assume its own conclusion. The proof is essentially complete modulo standard analytic number theory, and the alternative smoothing argument in the appendix is a useful contribution in itself. I regard the main theorem as very plausible, but the printed text has a load-bearing inconsistency in the definition of the truncation parameter y, and one imported uniform estimate needs verification.
major comments (2)
- [§4, Proposition 4.1 and §6, Proposition 6.1] Both propositions define y := x e^{-U}, but every subsequent estimate requires y = x^{e^{-U}}. With the printed definition, the assertion 'p≤y implies u_p ≥ U' is false; Ψ(x,y)/x does not tend to ρ(e^U) (it tends to ρ(1)=1); and the arithmetic tail bound in Proposition 6.1 becomes O(1) rather than O(e^{-U}). Specifically, in Proposition 4.1 Case 1 the sentence 'Since x=y e^U' is consistent with the printed y, but the following Dickman limit and the kernel-tail estimate require log y = e^{-U} log x, i.e. y = x^{e^{-U}}; in Proposition 6.1 the displayed equality log y/log x = e^{-U} is false for y = x e^{-U}. This invalidates the U→∞ step in the two convolution formulae. The error is repairable by replacing the definition with y := x^{e^{-U}} and adjusting 'x=y e^U' to 'x=y^{e^U}', but as written it is a load-bearing gap.
- [§2.3, Lemma 2.7 and §6, Proposition 6.1] The small-p tail bound in Proposition 6.1 relies on Lemma 2.7, which is stated uniformly for 2≤y≤x but is not proved; it is quoted from Tenenbaum [7] and Alladi–Johnson [2, Theorem 6*]. The proof only needs y = x^{e^{-U}} with U≥2, and the conclusion Ψ2(x,y) ≪ x log y/log x = x e^{-U} requires the implied constant to be absolute, independent of U. The paper should either provide a proof of Lemma 2.7 in the needed range or give the precise statement and conditions of the cited result, confirming that the constant does not depend on U. As it stands, the O(M e^{-U}) error that drives the U→∞ limit rests on an unverified uniformity.
minor comments (4)
- [Title/header] The arXiv header has typographical errors: 'RIGIDITY OF A VERAGES' and 'F ACTORS' should read 'AVERAGES' and 'FACTORS'.
- [Abstract and §1.2] The phrase 'log log-scale' is used inconsistently; the standard spelling is 'log-log scale'.
- [§4, Proposition 4.1] After the correction of y, the sentence 'Since x=y e^U' must be changed to 'Since x=y^{e^U}' to match the Dickman limit ρ(e^U).
- [§2.3, Lemma 2.7] The reference to [7, (1.5)–(1.6)] and [2, Theorem 6*] would be more useful if it included the exact range of uniformity and the shape of the implied constant; currently the reader cannot verify the absolute-constant claim without consulting the cited papers.
Circularity Check
No circularity: the derivation is self-contained modulo standard analytic number theory and independent citations.
full rationale
The derivation chain is not circular. The paper constructs the measures Λ_f and Λ_{f,t} directly from primes and the bounded function f, with no fitted parameters. Proposition 4.1 derives A_1(X_t;f) as K_1 * Λ_f + o(1) using the Dickman–de Bruijn asymptotic (Lemma 2.1), Mertens estimates (Lemma 2.4), and a three-range splitting; Proposition 6.1 derives the analogous formula for A_2 using the imported uniform estimate Ψ_2(x,y) ≪ x log y / log x (Lemma 2.7), which is cited from Tenenbaum and Alladi–Johnson and is independent of the target theorem. The Tauberian step (Section 7) uses only ∫ K_1 = 1, the proved no-real-zero fact for the Fourier transform of K_1 (Proposition 7.2), and Wiener's Tauberian theorem as a standard external result. No input is defined in terms of the conclusion, no parameter is fitted to a subset of data and then called a prediction, and the only self-referential material (the acknowledgement of AI assistance and of Tenenbaum's comments) is not load-bearing. The textual inconsistency in the printed definition y = x e^{-U} (whereas every subsequent use requires y = x^{e^{-U}}) and the fact that Lemma 2.7 is cited rather than proved are correctness or rigor concerns, not circularity; they do not make the derivation equivalent to its inputs.
Assumptions & free parameters
assumptions (6)
- standard math Uniform Dickman-de Bruijn asymptotic (Lemma 2.1)
- standard math Mertens' estimates for reciprocal primes and log p/p
- standard math Crude smooth-number bound Ψ(x,y)≪x e^{-u/2} (Lemma 2.3)
- standard math Uniform bound Ψ2(x,y)≪x log y/log x (Lemma 2.7)
- standard math Wiener's Tauberian theorem (Theorem 7.3)
- standard math Chebyshev estimate π(x)≪x/log x
Cite this review
Pith. "Pith review of Rigidity of Averages over the Two Largest Prime Factors." pith.science (2026). https://pith.science/paper/SWJWBTN5
@misc{pith2026260805191,
author = {Pith},
title = {Pith review of: Rigidity of Averages over the Two Largest Prime Factors},
year = {2026},
howpublished = {\url{https://pith.science/paper/SWJWBTN5}},
note = {Machine review of arXiv:2608.05191}
}
abstract
Let \(P_1(n)\) and \(P_2(n)\) be the largest and second-largest distinct prime factors of \(n\), respectively. Alladi and Johnson asked whether there exists a bounded function \(f\) on the primes for which both limits \(\frac{1}{x}\sum_{2\le n\le x} f(P_1(n)) \longrightarrow \kappa_1\) and \(\frac{1}{x}\sum_{2\le n\le x} f(P_2(n)) \longrightarrow \kappa_2\) exist with \(\kappa_1\neq\kappa_2\), where we set \(f(P_2(n))=0\) when \(n\) is a prime power. We prove that this is impossible: convergence of the first average forces convergence of the second to the same limit. On the \(\log\log\)-scale, the two averages are expressed as convolutions with explicit Dickman kernels. The Fourier transform of the Dickman kernel associated with the \(P_1\)-average has no real zeros. Wiener's Tauberian theorem then yields vague convergence of the translated measures associated with the weighted prime sums. The convergence of the second average follows.
Reference graph
Works this paper leans on
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[6]
Y. Katznelson,An Introduction to Harmonic Analysis, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004
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[8]
G. Tenenbaum,Introduction to Analytic and Probabilistic Number Theory, 3rd ed., Graduate Studies in Mathematics, vol. 163, American Mathematical Society, Providence, RI, 2015. Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin, USA Email address:dchen426@wisc.edu
work page 2015
Reviewed August 7, 2026 · model on record in the stance chip above.
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