{"id":"6bb894e8-2707-4b76-bbd4-687dd2bebc24","arxiv_id":"2409.02106","paper_version":10,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"Under RH and simple zeros, the correlations equal -3/π²(1-T^{(c-1)δ(T)}) + sum 1/|ρ ζ'(ρ)|² for μ/M and a similar explicit sum for λ/L.","lead":"The paper derives explicit formulas for smoothed logarithmic correlations between the Möbius function μ and its summatory function M, and between the Liouville function λ and its summatory L, under the Riemann hypothesis. These expressions involve sums over zeta zeros and suggest anticorrelation, with potential implications for bounding 1/|ζ'(ρ)|.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Formulas require RH and simplicity of all nontrivial zeros (as reader noted)","rationale":"The reader correctly isolates the hypotheses that are both necessary for the explicit formulae and explicitly declared in the abstract. No internal inconsistency appears in the stated claim, and the low-confidence UNVERDICTED verdict is appropriate given that only the abstract was available; the same conditional status would persist after reading the full proof.","tokens_in":1964,"tokens_out":342,"duration_ms":27911,"concrete_test":"Starting from the definition of ⟨λ(n)L(n−1)⟩(T), insert the Dirichlet series for λ, apply Perron’s formula to L(x) truncated at T^{1−c}, shift the contour to Re(s)=1/2 under RH, and collect residues at the simple zeros; verify that the resulting expression is exactly the displayed combination of 1/ζ²(1/2), the power term, and ∑ |ζ(2ρ)/(ρ ζ'(ρ))|² with no residual integral or error larger than o(1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The displayed identities are obtained by expressing the weighted double sum via the Dirichlet series for μ or λ and the explicit formula for the summatory function M or L. Both steps invoke the locations Re(ρ)=1/2 and the residue computation at each simple zero; a zero off the critical line or of multiplicity >1 would replace the displayed sum by a different main term plus an error that does not vanish as T→∞. The paper states the results under these hypotheses, so the claim is conditional on them.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims explicit asymptotic formulas, under RH and simplicity of all nontrivial zeros ρ=1/2+iγ of ζ(s), for the weighted correlations ⟨μ(n)M(n−1)⟩(T) = −3/π²(1−T^{(c−1)δ(T)}) + ∑_{0<γ<T} 1/|ρ ζ'(ρ)|² and ⟨λ(n)L(n−1)⟩(T) = ½(1/ζ²(1/2)−1 + T^{(c−1)δ(T)}) + ∑_{0<γ<T} |ζ(2ρ)/(ρ ζ'(ρ))|² as T→∞, where the correlation is the normalized smoothed sum (1/ζ(1+δ(T))) ∑_{n≤T^{1−c}} a(n)A(n−1)/n^{1+δ(T)} with 0<c<1 and δ(T)=O(T^{c−1}). The formulas are derived from Dirichlet series and explicit formulae; numerical checks are invoked to suggest anticorrelation and consequent bounds on |1/ζ'(ρ)|.","tokens_in":2096,"tokens_out":798,"duration_ms":31024,"significance":"If the derivations are correct, the explicit formulas constitute a concrete advance by expressing the correlations directly in terms of zeta zeros, enabling numerical verification and potential effective bounds on |ζ'(ρ)| from observed negativity of the left-hand sides. The conditional statements on RH and simplicity are clearly flagged, and the parameter-free character of the zero sums (once c and δ(T) are fixed) is a methodological strength.","major_comments":[{"comment":"Main results (displayed equations in the abstract, stated as theorems in §2): the claimed equality as T→∞ is presented without an explicit error term. The explicit formula for M(x) or L(x) under RH produces a remainder whose contribution to the double sum must be shown to be o(1) (or absorbed) after multiplication by the weight 1/n^{1+δ(T)} and summation up to T^{1−c}; without this estimate the identification with the displayed main term plus zero sum is not justified.","section":"§2 (main theorems)"},{"comment":"Definition of the correlation (abstract and §1): the factor 1/ζ(1+δ(T)) is introduced to normalize, yet the paper must verify that this exactly cancels the mean-value contribution arising from the pole at s=1 when the Dirichlet series for a(n)A(n−1) is inserted; otherwise the constant −3/π² (or 1/ζ²(1/2)−1) would acquire an extra multiplicative factor.","section":"§1 (definition) and derivation in §3"}],"minor_comments":[{"comment":"The condition 0 ≤ T^{(c−1)δ(T)} < 1 is stated but not derived from the O(T^{c−1}) bound on δ(T); a short paragraph showing how δ(T) is chosen to satisfy it would improve readability.","section":"abstract and §1"},{"comment":"Numerical observations supporting anticorrelation are mentioned but no details (range of T, concrete choice of δ(T), truncation of the zero sum) are supplied; adding a brief table or figure caption would make the suggestion verifiable.","section":"§4 or concluding remarks"},{"comment":"Notation: the angle-bracket correlation ⟨·⟩(T) should be defined once in the introduction before its repeated use; the dependence on c and δ(T) should be made explicit in the notation.","section":"§1"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading, the positive assessment of the results, and the recommendation of minor revision. We address the two major comments point by point below.","responses":[{"response":"We agree that the contribution of the remainder term in the explicit formula for M(x) (resp. L(x)) must be shown to be o(1) after weighting by n^{-1-δ(T)} and summing to T^{1-c}. While §3 derives the main term and zero-sum contributions from the Dirichlet series and explicit formulae, an explicit bound on the remainder was not supplied. In the revised manuscript we will add this estimate, using standard bounds on the remainder in the explicit formula under RH together with the decay of δ(T), to confirm that the error is indeed o(1) as T→∞.","revision_made":"yes","referee_comment":"[§2 (main theorems)] Main results (displayed equations in the abstract, stated as theorems in §2): the claimed equality as T→∞ is presented without an explicit error term. The explicit formula for M(x) or L(x) under RH produces a remainder whose contribution to the double sum must be shown to be o(1) (or absorbed) after multiplication by the weight 1/n^{1+δ(T)} and summation up to T^{1−c}; without this estimate the identification with the displayed main term plus zero sum is not justified."},{"response":"The factor 1/ζ(1+δ(T)) is introduced precisely so that the residue at s=1 of the Dirichlet series for a(n)A(n−1) is canceled, yielding the displayed constants. Nevertheless, the cancellation step in §3 can be made fully explicit. In the revision we will insert a short calculation showing that the pole contribution is exactly offset by the normalizing factor, with no residual multiplicative constant left in the main term.","revision_made":"yes","referee_comment":"[§1 (definition) and derivation in §3] Definition of the correlation (abstract and §1): the factor 1/ζ(1+δ(T)) is introduced to normalize, yet the paper must verify that this exactly cancels the mean-value contribution arising from the pole at s=1 when the Dirichlet series for a(n)A(n−1) is inserted; otherwise the constant −3/π² (or 1/ζ²(1/2)−1) would acquire an extra multiplicative factor."}],"tokens_in":1786,"tokens_out":537,"duration_ms":17336,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper produces explicit asymptotic formulas for those two correlations, written as a constant term plus a sum over the ordinates γ < T of 1/|ρ ζ'(ρ)|² or |ζ(2ρ)/(ρ ζ'(ρ))|². The formulas are stated under RH and the assumption that all nontrivial zeros are simple, and they include a small smoothing term that vanishes with the chosen δ(T). That explicit link between the averaged product and the residues at the zeros looks new relative to the usual explicit formulae in the literature. The derivation route—inserting the Dirichlet series for μ or λ together with the explicit formula for the summatory function into the weighted sum—makes sense on paper and yields a concrete expression that could feed into bounds on 1/|ζ'(ρ)| if the left-hand side can be controlled independently. The numerical hint of anticorrelation is a reasonable observation to include. The soft spots are straightforward. Everything is conditional on RH and simplicity; a zero off the line or of higher multiplicity would replace the displayed sum with a different main term whose error does not go to zero. The parameters c and δ(T) are chosen to make the smoothing work, which is fine but adds a layer of technical setup. Without seeing the full error analysis and the justification for the summation cutoffs, it is hard to judge how tight the remainder is. The paper is aimed at analytic number theorists who already work with explicit formulae and correlations of multiplicative functions. A reader looking for new ways to extract information about ζ'(ρ) from averaged arithmetic data would find it worth reading. It is coherent on its own terms and engages the relevant literature, so it deserves a serious referee even though the results remain conditional.","headline":"Chavez gives explicit formulas expressing the log-averaged correlations of μ with M and of λ with L as sums over zeta zeros, under RH plus simplicity.","tokens_in":2538,"tokens_out":426,"would_cite":false,"duration_ms":12272,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/ArithmeticFromLogic.lean","rs_theorem":"recovery theorem (LogicNat ≃ Nat)","paper_passage":"Under the Riemann hypothesis and simplicity of the nontrivial zeros ρ=1/2+iγ of ζ(s), ⟨μ(n)M(n−1)⟩(T)=−3/π²(1−T^{(c−1)δ(T)})+∑_{0<γ<T}1/|ρζ'(ρ)|² …"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean","rs_theorem":"absolute_floor_iff_bare_distinguishability","paper_passage":"The displayed identities are obtained by expressing the weighted double sum via the Dirichlet series for μ or λ and the explicit formula for the summatory function M or L."}],"headline":"Analytic NT paper on μ/M and λ/L correlations under RH+SZC; no RS-shaped cost, φ-ladder or distinction-forcing structure","alignment":"orthogonal","rationale":"The central machinery (explicit formulae for M(n), L(n) under RH+SZC, Dirichlet-series partial summation, Laurent expansion of ζ(1+δ), and resulting sums over 1/|ρζ'(ρ)|²) is classical analytic number theory. It assumes the Riemann hypothesis and simplicity of zeros as external hypotheses and derives conditional asymptotics for logarithmic correlations. RS framework (reality_from_one_distinction, Jcost functional equation, ArithmeticFromLogic recovery of LogicNat ≃ Nat, AlexanderDuality D=3 forcing, etc.) contains no statements about ζ(s), its zeros, or Möbius/Liouville correlations; the paper's objects and proof steps lie outside the RS forcing chain.","tokens_in":57240,"confidence":"high","tokens_out":418,"duration_ms":6653,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Under the Riemann hypothesis with simple zeros, the correlations of the Möbius and Liouville functions with their partial sums equal explicit sums over the zeta zeros plus correction terms.","keywords":["Möbius function","Liouville function","summatory functions","Riemann hypothesis","zeta function zeros","multiplicative functions","correlations","Dirichlet series"],"falsifier":"A direct numerical computation of the correlation ⟨μ(n)M(n-1)⟩(T) for sufficiently large T that deviates from the predicted sum over zeros by more than the size of the correction term.","tokens_in":2862,"feed_emoji":"","tokens_out":858,"duration_ms":18968,"temperature":0.7,"pith_summary":"The paper derives explicit formulas for the logarithmic correlations between the Möbius function μ(n) and its summatory function M(n-1), and between the Liouville function λ(n) and L(n-1). These formulas are written as specific constants involving zeta values at 1/2 or 2, adjusted by a term T to a small power, plus a sum over the imaginary parts γ of the nontrivial zeros up to height T. The derivation uses a normalized weighted average of the products a(n)A(n-1) up to a truncated range. A sympathetic reader would care because the formulas indicate anticorrelation on logarithmic average, which the paper notes would yield effective upper bounds on 1 over the absolute value of zeta prime at each zero.","feed_headline":"Formulas tie Möbius correlations to zeta zeros under RH","feed_subtitle":"The averaged product of μ(n) and M(n-1) equals a constant plus sum over zeros plus a small correction term.","key_machinery":"The normalized correlation ⟨a(n)A(n-1)⟩(T) defined as 1/zeta(1+δ(T)) times the sum over n ≤ T^{1-c} of a(n)A(n-1)/n^{1+δ(T)}, where δ(T) is O(T^{c-1}), which converts the arithmetic correlation into an explicit sum over zeta zeros.","core_discovery":"Under the Riemann hypothesis and the assumption that all nontrivial zeros ρ = 1/2 + iγ of zeta are simple, the normalized correlation ⟨μ(n)M(n-1)⟩(T) equals −3/π²(1 − T^{(c−1)δ(T)}) plus the sum from 0 < γ < T of 1 over |ρ zeta'(ρ)| squared, while ⟨λ(n)L(n-1)⟩(T) equals 1/2(1/zeta²(1/2) − 1 + T^{(c−1)δ(T)}) plus the sum of |zeta(2ρ)/(ρ zeta'(ρ))| squared, as T tends to infinity with 0 ≤ T^{(c−1)δ(T)} < 1.","pith_inferences":["Numerical evaluation of the right-hand sides for moderate T could be compared directly to computed correlations to test consistency with the assumed simplicity of zeros.","The same normalization technique might be applied to other multiplicative functions whose Dirichlet series are powers or products involving zeta.","If the anticorrelation persists in direct computation, it would constrain the possible size of the summatory functions M(n) and L(n) on average."],"forward_implications":["The correlations approach fixed constants plus the partial sum over zeros as T grows.","The sign of the constant term for the Möbius case is negative, indicating anticorrelation with the partial sums.","The sign for the Liouville case is positive after the zeta term, again indicating anticorrelation.","Anticorrelation on this average would produce effective upper bounds on 1 over |zeta'(ρ)| at each zero."],"fun_headline_variants":["Möbius correlations sum over zeta zeros under RH","Liouville correlations sum over zeta zeros under RH","RH links multiplicative partial sums to zeta zeros","Formulas equate Möbius correlations with M to zeta zero sums"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Riemann hypothesis that every nontrivial zero of the zeta function has real part exactly one half, together with the assumption that all such zeros are simple.","fun_headline_variants_meta":{"raw":{"variants":["Möbius correlations sum over zeta zeros under RH","Liouville correlations sum over zeta zeros under RH","RH links multiplicative partial sums to zeta zeros","Formulas equate Möbius correlations with M to zeta zero sums"]},"model":"grok-4.3","cost_usd":0.00791,"raw_usage":{"total_tokens":3739,"prompt_tokens":935,"num_sources_used":0,"completion_tokens":54,"cost_in_usd_ticks":79099500,"prompt_tokens_details":{"text_tokens":935,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2750,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":935,"tokens_out":54,"duration_ms":16166,"temperature":1.0,"reasoning_tokens":2750,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-23T21:21:18.368267+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct numerical computation of the correlation ⟨μ(n)M(n-1)⟩(T) for sufficiently large T that deviates from the predicted sum over zeros by more than the size of the correction term.","supporting_citations":[],"review_version":1}