{"id":"a0acb838-0d62-40e2-96c2-ce0feeb1e7cb","arxiv_id":"2607.25002","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"RH is equivalent to subpolynomial growth of arbitrarily high finite local moments of normalized Möbius polynomials on arcs of radius c/N.","lead":"The paper proves that the Riemann hypothesis is equivalent to subpolynomial growth of high local moments of Möbius Fourier polynomials sampled on arcs of width 1/N. It gives a clean critical-scale probabilistic criterion that recovers the Mertens function from local Lq data, without claiming a proof of RH.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No significant objection identified: the equivalence proof is elementary and every step checks out; the reader correctly located the only soft spot (the crude persistence bound), which is a limitation on sharpness, not a correctness gap.","rationale":"The reader's verdict (ACCEPT, high confidence, low correctness risk) matches my independent re-derivation of every computational step in the paper. The argument is short, self-contained, and uses only elementary tools (mean value theorem, partial summation, the classical Mertens–RH equivalence), and each piece survives line-by-line checking with constants and normalizations intact. The reader's identified weakest assumption is indeed the right place to look: Lemma 3.1 uses only the trivial derivative bound, which is what forces arbitrarily high moments in the converse. But this is a statement about the efficiency of the recovery mechanism, not a hole in it — the max-structure of Proposition 3.2 absorbs both regimes, and the q→∞ loss-erasure is rigorous. On significance, I concur with the reader: the forward direction shows that on the critical arc RH gives essentially pointwise subpolynomial control, so the local-moment hypothesis is morally the pointwise bound restated in L^q form, and the converse recovers only the single value P_N(0). The paper states this limitation plainly (\"not presented as a method for resolving RH itself\"), so there is no overselling to correct. Minor non-substantive nits: the proof of Proposition 3.2 cites \"Theorem 3.1\" where Lemma 3.1 is meant, and the acknowledged LLM proofreading does not touch the mathematics. Neither affects the verdict. Honest non-finding: no load-bearing concern; ACCEPT stands unchanged.","tokens_in":9575,"tokens_out":3967,"duration_ms":130007,"concrete_test":"Numerical sanity check of the central inequality: for N = 10^3,...,10^6, c = 1, q ∈ {2,4,8}, compute M(N) exactly and M_{q,c}(N) by fine quadrature of |P_N| on [−1/N, 1/N], then verify Corollary 3.3 holds with C_{q,c} = max{2, C_q c^{1/(q+1)}} as stated. A violation (e.g., from a dropped factor of 2, π, or √N in (6) or (7)) would appear immediately; agreement across all (N,q) confirms the normalization chain. Additionally, verify Lemma 3.1's persistence window empirically at one N where |M(N)| is large.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I re-verified the load-bearing chain of Theorem 1.3 and could not find a correctness concern that lands. (1) Lemma 3.1: |S'_N(t)| ≤ 2πΣn = πN(N+1) from |a_n|≤1 is correct, and the persistence conclusion |S_N(t)| ≥ A/2 for |t| ≤ A/(2πN(N+1)) follows by the mean-value estimate. (2) Proposition 3.2: both regimes of A/(2L) vs. r are handled; the algebra A^{q+1} ≤ 2^{q+1}Lr·B_q^q with L ≤ 2πN² gives C_q = (2^{q+2}π)^{1/(q+1)}, matching the statement. (3) Identity (6): B_q(S_N; c/N) = √N·M_{q,c}(N) checks out (1/(2r) = N/(2c), |S_N|^q = N^{q/2}|P_N|^q). (4) Corollary 3.3's exponent 1/2 + 1/(2(q+1)) is correct: (cN)^{1/(q+1)}·N^{q/(2(q+1))} = N^{(q+2)/(2(q+1))}. (5) Forward direction: the partial-summation identity (8) has the correct sign and boundary behavior (M(y)=0 for y<1), yielding |S_N(t)| ≪_δ N^{1/2+δ}(1+N|t|), hence |P_N| ≪ N^δ on the arc, giving all moments at once. (6) Converse: the order of quantifiers in (iii)⇒(i) is handled correctly — η is fixed first (η<ε/3), then q∈Q_η chosen large (1/(2(q+1))<ε/3), so both terms of (7) are O(N^{1/2+ε}), and constants depending on fixed (η,q,c) are harmless. The classical Mertens equivalence then gives RH. The reader's flagged weakest assumption (the trivial derivative bound forcing the 1/(2(q+1)) loss) is real but is openly acknowledged in Remark 3.4 and the introduction; it limits the tool's sharpness, not the theorem's truth. The paper's own framing — an equivalent reformulation, explicitly not an approach to proving RH — is accurate and not oversold.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies the normalized Möbius Fourier polynomial P_N(t) = N^{-1/2} Σ_{n≤N} μ(n) e^{2πint}, evaluated at a uniform random point U_{N,c} in the arc [-c/N, c/N]. The main result, Theorem 1.3, states that for fixed c > 0 the Riemann hypothesis is equivalent to subpolynomial growth M_{q,c}(N) = O_{η,q,c}(N^η) of the local moments for every η > 0 and every finite q ≥ 1 (and to an a priori weaker version in which, for each η, the bound is only required along an unbounded set of exponents Q_η). The forward direction uses the classical Mertens form of RH and partial summation (identity (8)); the converse uses a local moment-to-point-value inequality (Proposition 3.2), based on the derivative bound |S'_N| ≤ πN(N+1), which recovers |M(N)| from local L^q-data with a loss of N^{1/(2(q+1))} (Corollary 3.3). The paper also discusses the naturality of the critical scale N^{-1}, tail/Orlicz variants, and the relation to flatness and semiflatness questions for trigonometric polynomials.","tokens_in":10106,"tokens_out":2352,"duration_ms":77710,"significance":"The paper is explicitly and honestly framed: it does not claim a new approach to proving RH, but an equivalent local probabilistic reformulation, together with a quantitative mechanism (the moment-to-point-value inequality (5) and its corollary (7)) showing how high local moments on the critical-scale arc recover the Mertens function. Within that scope the paper delivers what it promises. I verified the load-bearing steps: Lemma 3.1 and the persistence argument; both regimes in Proposition 3.2, including the constant C_q = (2^{q+2}π)^{1/(q+1)}; the normalization identity (6); the exponent 1/2 + 1/(2(q+1)) in (7); the partial-summation identity (8) with the correct boundary behavior; and the quantifier order in the (iii)⇒(i) argument (η fixed first, then q ∈ Q_η chosen large), which is handled correctly and is the only place where the proof could easily have gone wrong. Constants are tracked explicitly and no unproved arithmetic input is used beyond the classical Mertens equivalence. The significance is modest by design — an equivalence with RH via an elementary inequality is a reformulation, not progress toward a proof — but the critical-scale local viewpoint, the explicit loss 1/(","major_comments":[],"minor_comments":[{"comment":"Cross-references are mislabeled throughout: the proof of Proposition 3.2 cites \"Theorem 3.1\" (should be Lemma 3.1); the proof of Corollary 3.3 cites \"Theorem 3.2\" (should be Proposition 3.2); Section 4 cites \"the precise form of Theorem 1.4\" (should be Remark 1.4); Section 10 cites \"Theorem 5.3\" (should be Question 5.3). Please correct all four.","section":"Sections 3, 4, 10"},{"comment":"Section 6: the tail-bound criterion is stated with the hedge \"together with a mild truncation or integrability condition.\" Since the trivial bound |X_{N,c}| ≤ √N always holds, the tail estimate P(|X_{N,c}| > λ) ≤ C_{q,η,c} N^{qη} λ^{-q} integrates directly (via E|X|^q = q∫_0^{√N} λ^{q-1} P(|X|>λ) dλ) to give M_{q,c}(N) ≪ N^η up to a logarithm, which Theorem 1.3 absorbs. Making this explicit would remove the vagueness and strengthen the section.","section":"Section 6"},{"comment":"Section 4, converse: the final exponent in the second term of (7) is 1/2 + 1/(2(q+1)) + ηq/(q+1); with η < ε/3 and 1/(2(q+1)) < ε/3 this is ≤ 1/2 + 2ε/3, so the conclusion O(N^{1/2+ε}) holds with room to spare. One line spelling out this arithmetic (rather than asserting it) would help the reader.","section":"Section 4, proof of Theorem 1.3"},{"comment":"The math italic 'e' for the exponential (e2πint in the extracted text) and the rendering of Möbius accents appear garbled in places; please check the compiled PDF for consistent typesetting of e^{2πint}, Mq,c(N), and the name Möbius throughout.","section":"Global/notation"},{"comment":"Remark 1.1 (the case c > N) is somewhat redundant with the sentence immediately preceding it (\"The finitely many values N ≤ 2c play no role...\"); the second paragraph on varying c_N is useful, but the first could be trimmed.","section":"Section 1, Remark 1.1"},{"comment":"The discussion of el Abdalaoui [1] in Sections 1 and 9 would benefit from one sentence stating precisely how the global semiflatness criterion's hypothesis compares quantitatively with (ii) of Theorem 1.3 (whole-circle Haar measure vs. shrinking-arc normalized measure), since this is the closest related result in the literature.","section":"Sections 1 and 9"}],"recommendation":"minor_revision","confidential_remarks":"The mathematics is elementary and, as far as I can verify, correct; the contribution is a cleanly proved equivalent reformulation of RH rather than progress toward it, so the editor should weigh whether the journal's bar for novelty is met by a well-executed criterion of this type. The citation pattern is unproblematic (self-citations [12, 15] are motivational only). The explicit acknowledgment of LLM assistance in proofreading and calculation-checking is transparent and appropriate; the mathematical content appears to be the author's."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that Verjovsky gives a clean, checkable equivalence: RH is exactly the statement that the normalized Möbius polynomials P_N, sampled uniformly on an arc of length 2c/N, have subpolynomial L^q moments for every finite q (or even just along unbounded sets of q depending on the exponent). The forward direction is partial summation plus the classical Mertens form; the converse is a new local moment-to-point-value inequality that recovers |M(N)| from those moments, with a 1/(2(q+1)) loss that vanishes as q grows.\n\nWhat is actually new is Proposition 3.2 / Corollary 3.3 and the critical-scale packaging in Theorem 1.3. The inequality is elementary (mean-value persistence from the crude derivative bound |S'_N| ≤ πN(N+1)), but it is not in the Denjoy–Kahane or el Abdalaoui global L^p-semiflatness literature the paper cites. The writing is careful about quantifiers in the weaker condition (iii), constants are tracked, and the paper does not oversell: it repeatedly says this is a reformulation, not a route to proving RH. Self-citations are background only. The math checks out end-to-end; the stress-test found no load-bearing error, and neither did I.\n\nThe soft spot is real but openly acknowledged and proportionate: the persistence estimate uses only |a_n| ≤ 1, so a single fixed moment only gets you an exponent strictly larger than 1/2. You need arbitrarily high moments to reach the critical line. That is a limitation on sharpness, not on truth. The maximal-scale question (how large can r_N be while RH still forces the moments) is left open and is the natural next question.\n\nThis is for people already working on probabilistic or flatness criteria for arithmetic Fourier polynomials. It will not reorganize analytic number theory, but it is a solid short note that belongs in the conversation. I would send it to referees without hesitation.","headline":"Sound elementary equivalence: RH iff subpolynomial local moments of Möbius polynomials on the critical arc of radius c/N; new local moment-to-point inequality, no correctness gap.","tokens_in":11363,"tokens_out":521,"would_cite":false,"duration_ms":8906,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N37","42A05","60G50"],"pacs":[],"model":"grok-4.5","headline":"The Riemann hypothesis is equivalent to subpolynomial growth of high local moments of Möbius Fourier polynomials on arcs of length 1/N.","keywords":["Möbius function","Mertens function","Riemann hypothesis","random Fourier series","local moments","trigonometric polynomials"],"falsifier":"Fix c>0 and check whether the local q-moments of P_N on [-c/N,c/N] remain O(N^η) for every η>0 as q is taken larger and larger; if for some fixed η the moments eventually grow faster than N^η no matter how large q is chosen, the claimed equivalence fails.","tokens_in":10894,"feed_emoji":"∫","tokens_out":1101,"duration_ms":14707,"temperature":0.7,"pith_summary":"This paper rewrites the classical Mertens form of the Riemann hypothesis as a statement about local moments. Form the normalized trigonometric polynomial whose coefficients are the Möbius values, then evaluate it at a point chosen uniformly at random inside an arc of radius c/N centered at the origin. The claim is that RH holds if and only if every finite moment of that random variable grows slower than every positive power of N. The reason the reformulation works is a quantitative inequality: a large value of the polynomial at zero cannot disappear instantly, so it remains visible to high enough local L^q averages on the critical scale set by the highest frequency. The paper does not offer a new attack on RH; it supplies an equivalent local probabilistic criterion that sits next to Denjoy's random-sign heuristic and global flatness criteria for the same polynomials.","feed_headline":"RH equals subpolynomial local moments of Möbius polynomials","feed_subtitle":"High moments on an arc of length 1/N recover the Mertens function and match the classical criterion","key_machinery":"The local moment-to-point-value inequality (Proposition 3.2 / Corollary 3.3): from the crude derivative bound |S'_N|≤πN(N+1) it recovers |M(N)| from the local L^q mean of S_N on an interval of length ~1/N, at the cost of a factor N^{1/(2(q+1))} that vanishes only as q\to∞.","core_discovery":"For any fixed c>0, the Riemann hypothesis is equivalent to the assertion that the local moments M_{q,c}(N) of the normalized Möbius polynomial P_N, sampled uniformly on the arc of radius c/N, satisfy M_{q,c}(N)=O(N^η) for every η>0 and every finite q≥1 (and also to the weaker version that only requires the bound along an unbounded set of exponents for each η). In short, subpolynomial growth of arbitrarily high finite local moments recovers the Mertens bound.","pith_inferences":["If a sharper persistence estimate that exploits multiplicative structure of µ could replace the crude derivative bound, a single fixed local moment might already be equivalent to RH.","The open maximal-scale question links the criterion directly to the size of additively twisted Möbius sums beyond the untwisted Mertens bound.","Comparing the growth of these local moments against the corresponding moments for independent random signs would give a quantitative measure of how much dependence the true Möbius coefficients retain at scale 1/N."],"forward_implications":["RH is equivalent to a critical-scale local moment condition on deterministic Möbius polynomials, with randomness only in the evaluation point.","A single fixed moment yields only an exponent strictly larger than 1/2; arbitrarily high moments are required to reach the classical Mertens threshold.","The same local-moment bound under RH extends at once to any shrinking radius r_N with N r_N = N^{o(1)}.","An Orlicz/sub-Gaussian strengthening of the local moments would still imply RH, giving a deterministic analogue of random-sign Fourier behaviour.","The maximal radius on which RH alone forces subpolynomial local moments is left open and is conjecturally no larger than N^{-1+o(1)}."],"fun_headline_variants":["RH equals subpolynomial local moments of Möbius polynomials","Local Möbius moments on 1/N arcs recover the Mertens bound","RH via subpolynomial growth of finite local P_N moments","Critical-scale local moments of Möbius polynomials match RH","High local moments of P_N on tiny arcs equivalent to RH"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The recovery step uses only the general bound on the derivative of a trigonometric polynomial with coefficients at most 1 in size; that forces the loss term that disappears only when moments of arbitrarily high order are allowed.","fun_headline_variants_meta":{"raw":{"variants":["RH equals subpolynomial local moments of Möbius polynomials","Local Möbius moments on 1/N arcs recover the Mertens bound","RH via subpolynomial growth of finite local P_N moments","Critical-scale local moments of Möbius polynomials match RH","High local moments of P_N on tiny arcs equivalent to RH"]},"model":"grok-4.5","effort":"low","cost_usd":0.00377,"raw_usage":{"total_tokens":1186,"prompt_tokens":782,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":37704000,"prompt_tokens_details":{"text_tokens":782,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":330,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":782,"tokens_out":74,"duration_ms":5473,"temperature":1.0,"reasoning_tokens":330,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T03:53:28.431809+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Fix c>0 and check whether the local q-moments of P_N on [-c/N,c/N] remain O(N^η) for every η>0 as q is taken larger and larger; if for some fixed η the moments eventually grow faster than N^η no matter how large q is chosen, the claimed equivalence fails.","supporting_citations":[],"review_version":1}