{"id":"a6a46233-080a-4659-af88-e9bbf525407c","arxiv_id":"2608.05191","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Convergence of the average of f(P1(n)) forces convergence of the average of f(P2(n)) to the same limit for every bounded f.","lead":"The paper proves that if the average of a bounded function of the largest prime factor of n converges, then the average of the same function of the second-largest prime factor converges to the same value. This settles a question posed by Alladi and Johnson about whether the two averages could converge to different limits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The small-prime truncation is printed inconsistently (y=x e^{-U} instead of y=x^{e^{-U}}) and the imported P2 tail estimate is unproved in the needed range; both are repairable, so the reader's CONDITIONAL verdict stands.","rationale":"The proof is structurally sound: the convolution identities are coherent, the no-real-zeros proof of Proposition 7.2 is correct, Lemma 7.1 handles the jump of K1 properly, and the Wiener Tauberian step is standard. I did not find a hidden flaw in the Fourier argument or in the passage from vague convergence to the kernel integrals. The load-bearing weak point is the small-prime tail in the two convolution propositions. As printed, the truncation y=x e^{-U} makes the displayed estimates false: for example, p≤y then gives u_p of order U/log x rather than U, so the kernel-tail sum is not O(e^{-U}). This is a fixable typographical error, but it invalidates Propositions 4.1 and 6.1 as submitted. The second issue is Lemma 2.7: it is imported without proof, and the proof needs the estimate only for y=x^{e^{-U}}, a range not stated with a verifiable constant. If the implied constant depends on U, the small-p contribution in Proposition 6.1 would not tend to zero after letting U→∞. Both concerns are consistent with the reader's CONDITIONAL assessment and do not change the verdict.","tokens_in":16273,"tokens_out":28958,"duration_ms":329566,"concrete_test":"Recompute Propositions 4.1 and 6.1 with the corrected definition y:=x^{e^{-U}}, checking that the tail estimates become O_M(e^{-U}) with constants independent of U and x; then prove or locate in [7] the uniform estimate Ψ2(x,y)≪x log y/log x for y=x^{e^{-U}}, U≥2, with an explicit absolute constant. If the constant grows with U, test whether C(U)e^{-U}→0; passing both steps removes the objection.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Propositions 4.1 and 6.1 both define y:=x e^{-U}, but every subsequent use requires y=x^{e^{-U}}: p≤y gives u_p≥U only for the latter, x=y^{e^U} rather than x=y e^U, and log y/log x=e^{-U}. With the printed definition, the claimed O(e^{-U}) tail bounds become O(1), so the U→∞ step in the two convolution formulas does not follow as written. Separately, Lemma 2.7 is imported from [7]/[2] and stated for all 2≤y≤x, whereas the proof only needs y=x^{e^{-U}}, U≥2; if the constant in the cited uniform estimate depends on U, then the small-p arithmetic tail M e^{-U} in Proposition 6.1 would not vanish after U→∞. No proof or precise constant/range is supplied. These are fixable, but until corrected the central Tauberian argument rests on an invalid truncation and an unverified uniform estimate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16439,"tokens_out":7392,"duration_ms":93598,"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":[{"comment":"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.","section":"§4, Proposition 4.1 and §6, Proposition 6.1"},{"comment":"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.","section":"§2.3, Lemma 2.7 and §6, Proposition 6.1"}],"minor_comments":[{"comment":"The arXiv header has typographical errors: 'RIGIDITY OF A VERAGES' and 'F ACTORS' should read 'AVERAGES' and 'FACTORS'.","section":"Title/header"},{"comment":"The phrase 'log log-scale' is used inconsistently; the standard spelling is 'log-log scale'.","section":"Abstract and §1.2"},{"comment":"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).","section":"§4, Proposition 4.1"},{"comment":"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.","section":"§2.3, Lemma 2.7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be genuinely novel and within the scope of the journal. The main theorem is likely correct, and the Fourier no-zero proof is a strong point. However, the printed truncation parameter in Propositions 4.1 and 6.1 is inconsistent with its usage, creating a genuine gap in the tail estimates; the fix is straightforward but must be checked carefully. I would also ask the author to supply a proof or a precise reference for Lemma 2.7 in the needed range. These are repairable, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. Chen settles the Alladi–Johnson question: convergence of the P1 average forces convergence of the P2 average to the same limit. The proof is built from explicit Dickman-kernel convolutions and a Wiener Tauberian step, and the Fourier no-zero lemma for K1 is proven directly from the Dickman equation—that's the real new ingredient. The normalizations check out; kernel masses integrate to 1 via the Dickman differential equation. The alternative smoothing proof in the appendix is a nice bonus and shows the author thought about robustness.\n\nThe soft spot is real but textual. Both Proposition 4.1 and 6.1 define y := x e^{-U}. Every use needs y = x^{e^{-U}}. With the printed definition, p≤y does not imply u_p ≥ U (it gives u_p = log log x - log log(x e^{-U}) ~ U/log x), and x = y e^U is not x = y^{e^U}. So the claimed O(e^{-U}) tails become O(1), and the U→∞ step fails. This is exactly the kind of typo that would be caught in revision; the surrounding text and the case analysis make the intended definition unambiguous, and fixing the exponent repairs the proof. I do not see a deeper flaw.\n\nThe imported Lemma 2.7 (Ψ2(x,y) ≪ x log y / log x) is load-bearing for the small-p tail in Proposition 6.1. It is cited to Tenenbaum and Alladi–Johnson, and the stated range 2≤y≤x is what their theorems give, so the citation pattern is clean. A referee should ask the author to state the constant's uniformity in the proof, but this is standard.\n\nMy verdict matches the reader's: conditional. The central claim is very likely correct, the framework is reusable, and the paper deserves a serious referee. If the author fixes the truncation definition and adds a line about Lemma 2.7, this is publishable.","headline":"A clean and likely correct resolution of the Alladi–Johnson rigidity question, held back only by a fixable truncation typo in the convolution propositions.","tokens_in":16989,"tokens_out":2648,"would_cite":true,"duration_ms":42767,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11N25","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"If the average of $f(P_1(n))$ converges, then the average of $f(P_2(n))$ converges to the same limit.","keywords":["largest prime factor","second-largest prime factor","Dickman function","Wiener Tauberian theorem","prime factor averages","log-log scale","vague convergence","rigidity"],"falsifier":"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.","tokens_in":16015,"feed_emoji":"🔢","tokens_out":9078,"duration_ms":97254,"temperature":0.7,"pith_summary":"This paper answers an open question of Alladi and Johnson by proving that no bounded function on the primes can make the averages over the largest and second-largest distinct prime factors converge to different limits. The main theorem states that if \\(A_1(x;f) = \\frac{1}{x}\\sum_{n\\le x} f(P_1(n))\\) converges to \\(\\kappa\\), then \\(A_2(x;f)\\), defined with \\(f(P_2(n))=0\\) at prime powers, also converges to \\(\\kappa\\). The proof works on the additive \\(\\log\\log\\)-scale, where both averages become convolutions with explicit kernels built from the Dickman function. Because the Fourier transform of the kernel for the first average has no real zero, Wiener's Tauberian theorem forces the relevant weighted prime measure to converge to \\(\\kappa\\) times Lebesgue measure, which then determines the second average.","feed_headline":"The largest prime factor fixes the second-largest prime factor average","feed_subtitle":"If the average over the largest prime factor converges, the second-largest average converges to the same limit.","key_machinery":"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