{"id":"4c63005e-bccc-461f-a96a-7779cbc3a5a2","arxiv_id":"2607.09110","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted averages of diagonal Möbius-twisted Elliott-Halberstam achieve the conjectured size under GRH and weak Gonek-Hejhal for Sobolev and Hölder-Zygmund weights.","lead":"Under GRH and a weak Gonek-Hejhal hypothesis, weighted averages of the diagonal Möbius-twisted Elliott-Halberstam discrepancy match the size expected by the diagonal conjecture, for all levels of distribution when weights are Sobolev W^{2,1} and for levels above roughly 1/2 when they are Hölder-Zygmund. The same diagonal form already implies binary Goldbach once classical EH is supplied by Bombieri-Vinogradov.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the already-flagged Conjecture 19.","rationale":"The central claim is a conditional upper bound of the expected size for a weighted diagonal average of the Möbius-twisted Elliott-Halberstam discrepancy. The argument is a careful but standard combination of GRH-explicit formulae, a two-dimensional Abel identity, and absolute-convergence estimates that rest on Conjecture 19. The reader already identified this conjecture as the sole soft point; no stronger or independent load-bearing flaw is present. The concrete test above merely quantifies how sensitive the final exponent is to a mild logarithmic weakening of that conjecture, which is the natural next check. Consequently the ACCEPT verdict and HIGH confidence stand.","tokens_in":40179,"tokens_out":448,"duration_ms":5363,"concrete_test":"Re-derive the main-term bound of Theorem 22 under the weaker hypothesis J_1(T) ≪ T (log T)^C for a fixed C>0; if the final exponent of N remains 2-ε (with only an extra log-power factor absorbed into E(f'')), the claim is robust to the known refinements of Gonek-Hejhal; if the exponent deteriorates below 2-ε the dependence is sharp.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly isolates the only load-bearing conditional input: Conjecture 19 (J_1(T) ≪ T). All other steps (two-dimensional Abel summation, truncated explicit formulae under GRH, absolute convergence of the double zero series via Theorem 15, and the resulting averaged bounds of Theorems 22/26/24/28) follow by standard estimates once that bound is granted. No internal inconsistency, hidden range restriction, or circularity appears in the derivation of the strongest claim. The paper already states the dependence on Conjecture 19 explicitly and uses only the weakest form needed for absolute convergence and error absorption.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces a diagonal variant (dEH^{μ,log}) of the Möbius-twisted Elliott–Halberstam conjecture and proves that it, together with a classical EH input of level θ, already implies the binary Goldbach conjecture (Theorem 7). It then establishes weighted averaged forms of this diagonal discrepancy. Under GRH and the weak Gonek–Hejhal bound J_1(T)≪T, for weights f with support in [0,β) belonging to the Sobolev space W^{2,1} the averaged sum over q≤N^{1-2ε} of (1/φ(q))|∑_{χ\neqχ_0} χ(N)∑∑ Λ(n)χ(n)μ(m)f((n+m)/N)| is ≪_ε N^{2-ε} E(f''), and an analogous bound holds for the logarithmically weighted sum; the same statements are obtained for Hölder–Zygmund weights in C^δ with a δ-dependent range of θ that never falls below 1/2-2ε. The proofs rely on a two-dimensional Abel summation identity, truncated explicit formulae for ψ(x,χ) and M(x) (extended to all x>0), absolute convergence of double series over zeros, and careful tracking of error terms.","tokens_in":40352,"tokens_out":998,"duration_ms":44329,"significance":"If the conditional results hold, they supply the first averaged evidence toward a diagonal form of the twisted Elliott–Halberstam conjecture that is already known to be strong enough for Goldbach. The two-dimensional Abel identity (Theorem 8) and its discrete counterpart cleanly decouple the arithmetic convolutions, while the absolute-convergence theorem for double zero series (Theorem 15) and the uniform explicit formulae for x>0 are reusable tools. The paper is careful to isolate the single non-standard hypothesis (Conjecture 19) and to show that the classical max_y and max_a can be removed from the Goldbach implication. These features make the work a solid, self-contained contribution to the analytic theory of Goldbach-type problems under standard hypotheses.","major_comments":[],"minor_comments":[{"comment":"The phrase “consistent with the bound of the diagonal versions” (abstract and §1.2) is slightly ambiguous: the proved upper bound is O(N^{2-ε}) while a true dEH would give O(N/log^A). A short remark comparing the GRH-trivial size, the size implied by dEH, and the size actually obtained would remove any possible misunderstanding.","section":null},{"comment":"Notation for the sum over characters is declared as an “abuse” (p. 5 and after (4.1)), yet the same symbol χ is used both for the non-principal character mod q and for its primitive inducer. A single clarifying sentence at the first occurrence would help the reader.","section":null},{"comment":"In several places (e.g., the statements of Theorems 21–28) the dependence of the implied constants on eta, δ, f is recorded only in the O-symbol; listing the parameters explicitly in the theorem statements would improve readability.","section":null},{"comment":"Minor typographical inconsistencies appear throughout (missing spaces after commas, occasional “log (N)” versus “log(N)”, and a few duplicated words such as “a verages”). A careful copy-edit pass is recommended.","section":null},{"comment":"The examples in §6 (Cesàro–Riesz and the Zygmund-type weight) are useful; it would be helpful to record the precise value of E(f'') or the Hölder norm for each example so that the reader can see the numerical size of the constant.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically solid and the conditional results are correctly derived. The only load-bearing external input is the weak Gonek–Hejhal bound, which is already stated clearly. I see no reason to doubt the correctness of the arguments once that hypothesis is granted. The paper is a natural fit for a number-theory journal that publishes conditional work on Goldbach and distribution of primes in arithmetic progressions."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper’s real contribution is a clean reduction of the full max-max Möbius-twisted Elliott–Halberstam to a pure diagonal form (dEH^{μ,log}) that still yields Goldbach when combined with classical EH, plus new weighted averages of that diagonal form under GRH and a weak form of Gonek–Hejhal (J_1(T) ≪ T). The two-dimensional Abel identity (Theorem 8) and its discrete counterpart let the author decouple the explicit formulae for ψ(x,χ) and M(x) (or the log-weighted version), then control the resulting convolutions with Sobolev W^{2,1} or Hölder–Zygmund C^δ weights. That machinery is new relative to Huang–Li and Murty–Vatwani, and it produces averaged bounds of the expected size N^{2-ε}E(f'') for the full range θ < 1-2ε when the weight is Sobolev, and for θ down to 1/2-2ε (or better when δ ≥ 3/2) in the Hölder–Zygmund case.\n\nThe technical work is careful. Explicit formulae are extended to all x > 0, double series over zeros are shown to converge absolutely by partial summation and Stirling (Theorem 15), and the error terms in Theorems 21–28 are tracked without circularity. The diagonal conjecture itself is only a mild reformulation, but the author proves it is still strong enough for Goldbach (Theorem 7), which justifies the whole enterprise. Cesàro–Riesz and Zygmund-type examples in Section 6 make the results immediately usable.\n\nThe only load-bearing soft spot is Conjecture 19. If the true growth of ∑ 1/|ζ'(ρ)|^{2} is larger by any positive power of log T, the error terms in the Sobolev theorems exceed the main-term size. Everything else is standard analytic number theory executed at a high level; the paper states the dependence explicitly and uses only the weakest form needed. No hidden range restrictions or internal inconsistencies appear.\n\nThis is for specialists working on parity-sensitive problems or weighted averages of arithmetic functions. It supplies reusable tools (the Abel identities and the averaged bounds) rather than a breakthrough on Goldbach itself. I would send it to a serious referee; the arguments are checkable line-by-line and the conditional progress is genuine.","headline":"Solid conditional averaged bounds for a diagonal Möbius-twisted EH that still implies Goldbach, under GRH plus a weak Gonek–Hejhal input.","tokens_in":40965,"tokens_out":601,"would_cite":true,"duration_ms":7020,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N37","11P32","11N56"],"pacs":[],"model":"grok-4.5","headline":"Under GRH and a weak zero-density hypothesis, weighted averages of the Möbius-twisted Elliott–Halberstam discrepancy match the expected diagonal size for all levels of distribution when the weight is Sobolev W^{2,1}.","keywords":["Elliott–Halberstam","Möbius function","binary Goldbach","explicit formulae","Sobolev weights","Hölder–Zygmund","Gonek–Hejhal","weighted averages"],"falsifier":"Compute or rigorously bound the first moment J_1(T)=∑_{0<γ≤T}1/|ζ′(ρ)|^{2}; if it exceeds T(log T)^c for every c>0, the error terms in the explicit formulae of Theorems 22 and 26 exceed N^{2−ε} and the averaged bounds fail.","tokens_in":41054,"feed_emoji":"🔢","tokens_out":835,"duration_ms":8982,"temperature":0.7,"pith_summary":"The paper studies averaged forms of the Möbius-twisted Elliott–Halberstam conjecture that appears in Huang–Li’s approach to binary Goldbach. Instead of the full maximal discrepancy, it considers smooth or Hölder–Zygmund weighted double sums of Λ(n)χ(n)μ(m) and the same sums with an extra log m. Under the generalised Riemann hypothesis and a weak form of the Gonek–Hejhal conjecture (the first moment of 1/|ζ′(ρ)|^{2} is O(T)), these averages are shown to be of the same size as the “diagonal” version of the conjecture. For weights whose second derivative is integrable the bound holds for every level of distribution up to N^{1−ε}; for Hölder–Zygmund weights of order δ the admissible level drops but never falls below N^{1/2−ε}. The same statements hold after the logarithmic weight is inserted. The author also records that the pure diagonal conjecture already implies Goldbach once classical Elliott–Halberstam is available, so the averaged results sit in a chain of implications that ends at the binary Goldbach problem.","feed_headline":"Weighted Möbius–EH averages hit expected size under GRH","feed_subtitle":"Sobolev weights give the full range of distribution levels; Hölder–Zygmund still stay above the square-root barrier","key_machinery":"A two-dimensional Abel summation identity that rewrites the weighted double sum as a Laplace convolution of the partial-sum functions of Λχ and μ; once the truncated explicit formulae for those summatory functions are inserted, the main term becomes a double series over zeros that can be controlled by Gamma-function estimates and the weak Gonek–Hejhal hypothesis.","core_discovery":"Under GRH and the bound J_1(T)≪T, the character-averaged weighted sum of Λ(n)χ(n)μ(m)f((n+m)/N) (and its logarithmic counterpart) is O_ε(N^{2−ε}E(f″)) for every Sobolev weight f∈W^{2,1} supported in [0,β), for all levels of distribution θ<1; the same size holds for Hölder–Zygmund weights of order δ with a θ that depends on δ but is always at least 1/2−2ε.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Sobolev weights give full θ-range Möbius-EH averages under GRH","Hölder-Zygmund weights keep Möbius-EH above √N barrier under GRH","Weighted Möbius-EH averages match diagonal size for all θ under GRH","GRH plus weak J1 bound yield expected weighted Möbius-EH size","Sobolev W^{2,1} weights achieve full distribution in Möbius-EH under GRH"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The assumption that the sum of 1/|ζ′(ρ)|^{2} over zeros up to height T grows at most linearly in T; any extra positive power of log T would make the error terms larger than the claimed main-term size.","fun_headline_variants_meta":{"raw":{"variants":["Sobolev weights give full θ-range Möbius-EH averages under GRH","Hölder-Zygmund weights keep Möbius-EH above √N barrier under GRH","Weighted Möbius-EH averages match diagonal size for all θ under GRH","GRH plus weak J1 bound yield expected weighted Möbius-EH size","Sobolev W^{2,1} weights achieve full distribution in Möbius-EH under GRH"]},"model":"grok-4.5","effort":"low","cost_usd":0.00774,"raw_usage":{"total_tokens":2006,"prompt_tokens":978,"num_sources_used":0,"completion_tokens":118,"cost_in_usd_ticks":77400000,"prompt_tokens_details":{"text_tokens":978,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":910,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":978,"tokens_out":118,"duration_ms":9690,"temperature":1.0,"reasoning_tokens":910,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T05:19:35.949541+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute or rigorously bound the first moment J_1(T)=∑_{0<γ≤T}1/|ζ′(ρ)|^{2}; if it exceeds T(log T)^c for every c>0, the error terms in the explicit formulae of Theorems 22 and 26 exceed N^{2−ε} and the averaged bounds fail.","supporting_citations":[],"review_version":1}