{"id":"bccc068a-07e2-4f72-b007-ae2285b46970","arxiv_id":"2607.09797","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If the Laplace-weighted Möbius sum Σ μ(n)e^{-nx} is O(x^{-1/2}) as x→0+, the Riemann hypothesis holds; an explicit formula with new double-pole terms is proved under a simplicity assumption.","lead":"The paper derives an exact formula for a weighted Möbius-function sum and shows that if that sum decays like the square root of its variable, the Riemann hypothesis follows. The formula contains a new double-pole, logarithmic term not present in the classical explicit formula for the Mertens function.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 1.3(a) is proved by a sound identity-theorem argument; the unproved hypotheses S/H affect only the explicit formula and the converse direction, exactly as disclosed.","rationale":"The reader's verdict was CONDITIONAL, centered on Hypothesis (S). My reading agrees that (S)/(H) are unproved conditions for the explicit formula and the converse, but the paper's headline claim as I understand it—Theorem 1.3(a)—does not rely on them. I re-derived the identity-theorem argument and found it complete. The only genuinely fragile point is the premise O(x^{-1/2}) itself, which is a strong regularity assumption of unknown truth; but an unproved hypothesis is not a proof gap. The paper is transparent about the asymmetry of the criterion. Hence no change to the conditional verdict.","tokens_in":23580,"tokens_out":26995,"duration_ms":238732,"concrete_test":"Although no objection is raised, one verification worth running is an independent high-precision numerical evaluation of the explicit formula (1.19) for x=1,10,100: compute the zero sum with, say, the first 10^4 zeros and the special functions κ,β,λ,υ from their defining series, and compare with the direct sum ∑ μ(n)e^{-nx}. Agreement to the full precision would independently corroborate the double-pole residues and the contour-limit argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the proof of Theorem 1.3(a) in detail, I find no load-bearing flaw in the central implication. Lemma 8.1 correctly gives G(s) holomorphic on Re(s)>1/2 from the O(x^{-1/2}) bound; F(s)=ζ(s)G(s)−Γ(s) is holomorphic on the connected domain D={Re(s)>1/2}\\{1}; the identity F=0 on Re(s)>1 extends by the identity theorem, and evaluating at a hypothetical zero ρ with Reρ>1/2 gives Γ(ρ)=0, impossible. The functional equation then excludes zeros left of 1/2. Step 2 carefully uses (1.5) only where proved. The O(x^{-1/2}) premise is unproved, but that is the hypothesis, not a gap. The paper's unproved Hypotheses (S) and (H) affect only the explicit formula and the converse Theorem 1.3(b), and this is disclosed explicitly (Remark 5.5, Section 7). I cannot identify an internal inconsistency in the residue computations or contour estimates.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the function Phi(e^{-x}) = sum_{n=1}^infty mu(n) e^{-nx}, whose Mellin transform is Gamma(s)/zeta(s). Under the assumption that all nontrivial zeros of zeta are simple (Hypothesis S), it derives an explicit formula expressing Phi(e^{-x}) as a sum over nontrivial zeros plus four explicitly defined entire functions kappa, beta, lambda, upsilon, with a logarithmic term -2 beta(x) log(2 pi x) that arises from the collision of poles of Gamma with the trivial zeros of zeta. It also proves absolutely convergent Mobius-Bessel series representations for kappa, beta, lambda. The central criterion is Theorem 1.3(a): if Phi(e^{-x}) = O(x^{-1/2}) as x -> 0+, then the Riemann hypothesis follows, with no hypothesis on the zeros of zeta. The converse direction, Theorem 1.3(b), is proved under RH together with Hypotheses S and H.","tokens_in":1097,"tokens_out":1734,"duration_ms":174271,"significance":"If correct, the explicit formula is a new structural result, distinguished from the classical Mertens explicit formula by the presence of double poles at the trivial zeros and the resulting logarithmic term. The unconditional direction of the RH criterion is a new sufficient condition of the Hardy-Littlewood-Riesz type. The paper's strengths are the detailed and transparent proof structure, the cross-checked residue computations in Sections 3-4 and Appendices D-H, and the non-circular identity-theorem argument in Theorem 1.3(a), which applies the identity theorem to zeta*G - Gamma rather than to 1/zeta. The main limitations - that the explicit formula is conditional on unproved Hypothesis S and the converse direction on unproved Hypothesis H - are explicitly disclosed, and Remark 5.5 gives an unconditional form of the explicit formula with multiplicity-adjusted zero terms. I found no load-bearing technical error in the proof of the central implication.","major_comments":[],"minor_comments":[{"comment":"The phrase \"all the zeros of the Riemann zeta function are simple\" should be qualified as \"all nontrivial zeros\", matching Hypothesis (S) in Section 1.1. The trivial zeros are simple and are not at issue in the conjecture.","section":"Abstract"},{"comment":"The lemma states the lower bound (5.14) for -1 <= sigma <= 2 and cites Titchmarsh Theorem 9.7. Since the standard theorem is often stated for 1/2 <= sigma <= 2, please add one sentence explaining how the extension to -1 <= sigma <= 1/2 follows from the functional equation, or confirm that the cited theorem already contains this range.","section":"Section 5.2, Lemma 5.2"},{"comment":"The remark that the first zero has imaginary part 14.134... is not used in the displayed pairing argument (7.3)-(7.9). It can be removed or explicitly connected to the claim that rho and 1-rho lie in opposite halves of the strip.","section":"Section 7"},{"comment":"The statement that the main identity was \"verified numerically to thirty significant digits at several values of x\" is not reproducible without the x-values and the computed values. Either supply the numerical data or soften the claim.","section":"Acknowledgements"},{"comment":"The labels T_N-1 and T_N-2 on the imaginary axis are unexplained; since the heights are supplied by Lemma 5.2, clarify that these are schematic markers, not all distinct ordinates used in the proof.","section":"Figure 1"},{"comment":"In the displayed bound, the constant C_2 absorbs the factor (2 pi x)^{-1/2} from (2 pi x)^{2N-1/2}. It would be clearer to define C_2 accordingly, or to keep (2 pi x)^{2N} and note the absorbed factor explicitly.","section":"Section 5.3, Eq. (5.19)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is technically sound as far as I have checked. The conditional status of the main formula and of the converse direction is openly disclosed, and the central unconditional implication is proved without circularity or hidden assumptions. The only requests are clarifications and minor presentation adjustments. The paper is a reasonable fit for the journal provided the minor issues are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is stronger than the math.GM tag suggests. The genuinely new item is the explicit formula (1.19): the Laplace kernel makes the poles of Gamma collide with the trivial zeros of zeta, producing double poles and the logarithmic term -2β(x)log(2πx). That structure is not in the classical Mertens formula or the Hardy-Littlewood/Riesz/Báez-Duarte family, and the residue computations in Sections 3-4 and Appendices D-H check out. The direct Laurent computation agreeing with the sine-splitting computation is a solid internal consistency check.\n\nThe second genuinely good piece is Theorem 1.3(a). The proof via ζ(s)G(s)-Γ(s) and the identity theorem is clean and non-circular. I agree with the stress-test note: the argument does not assume holomorphy of 1/ζ, and the O(x^{-1/2}) bound is the hypothesis, not a gap. The functional-equation step that rules out zeros left of the critical line is fine. This is a legitimate sufficient criterion for RH, even though it is one-way and the premise is a strong unproved regularity statement.\n\nThe soft spots are exactly the ones the reader flags. Hypothesis (S) is load-bearing for the explicit formula and for the converse Theorem 1.3(b); without it, you only get the multiplicity-adjusted form of Remark 5.5. The converse also needs Hypothesis (H), an extra condition on the zero sum. These are disclosed honestly, but they mean the headline identity is conditional. The numerical check mentioned in the acknowledgements is not reproducible as shipped; that is minor because the analytic proof is the main content, but I would ask for the script or at least a few more digits at multiple x values.\n\nThe citation pattern looks reasonable: the novelty claims against Titchmarsh, Bartz, Báez-Duarte, and the Mellin/Abel criterion literature are cautious and, as far as I can tell, accurate. The AI acknowledgment is transparent and does not change my assessment; the mathematics stands or falls on its own.\n\nThis deserves a serious referee, not a desk reject. If I were handling it, I would send it to someone who knows explicit formulas and ask them to check the contour estimates on the moving line and the novelty claim. My own verdict would be conditional acceptance, with the hypotheses kept prominent and the numerics reproduced. I would cite this if I worked on Möbius explicit formulas, and I would bring it to a reading group interested in RH criteria.","headline":"A sound, genuinely new explicit formula for the Laplace-kernel Möbius sum, plus a clean unconditional proof of one implication of an RH criterion; the main caveat is that the headline identity itself depends on an unproved simplicity hypothesis.","tokens_in":752,"tokens_out":1867,"would_cite":true,"duration_ms":35207,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","33C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A decay bound on the discrete Laplace transform of the Möbius function forces the Riemann hypothesis, and a new explicit formula exposes the mechanism.","keywords":["Riemann zeta function","Möbius function","Riemann hypothesis","explicit formula","Bessel functions","simple zeros","Mellin transform","discrete Laplace transform"],"falsifier":"Evaluate both sides of the explicit formula at x=0.1, x=0.5, and x=1 to 60 significant digits; any mismatch between the direct Möbius sum and the zero-plus-special-function side would falsify the residue computation. For the RH implication there is no numerical experiment that could directly falsify it—a counterexample would require RH to be false and the O(x^{-1/2}) bound to hold.","tokens_in":23464,"feed_emoji":"🧮","tokens_out":9570,"duration_ms":85007,"temperature":0.7,"pith_summary":"The paper's central aim is a criterion for the Riemann hypothesis from the discrete Laplace transform of the Möbius function, Φ(e^{-x}) = Σ μ(n)e^{-nx}. It proves that if Φ(e^{-x}) = O(x^{-1/2}) as x→0+, then every nontrivial zero of ζ lies on the critical line, with no hypothesis on the zeros needed for that direction. The same machinery, under the assumption that all nontrivial zeros are simple, yields an explicit formula for Φ(e^{-x}) as a zero sum plus entire functions, in which the collision of Gamma poles with trivial zeta zeros creates double poles and a logarithmic residue not present in the classical Mertens formula. Three of the resulting entire functions admit absolutely convergent closed forms as Möbius-weighted Bessel series. A sympathetic reader would care because the bound-to-RH direction turns a single asymptotic estimate into the full Riemann hypothesis.","feed_headline":"A decay bound on the Möbius sum forces the Riemann hypothesis","feed_subtitle":"An explicit formula exposes the mechanism: x^{-1/2} decay forces every nontrivial zero onto the critical line.","key_machinery":"The central object is the Mellin inversion formula Φ(e^{-x}) = (1/2πi)∫_{3/2-i∞}^{3/2+i∞} Γ(s)/ζ(s) x^{-s} ds. Shifting this contour left past the trivial zeros is the mechanism: residues at odd negative integers produce κ, residues at even negative integers are evaluated as double poles whose log term yields β, λ, and υ, and residues at nontrivial zeros give the ζ'(ρ)-weighted zero sum. Two estimates carry the contour shift: a uniform lower bound on |Γ(σ+it)| supplying exponential cancellation on the moving line Re(s)=1/2−2N, and a classical selection of horizontal heights avoiding zeros of ζ. For the Riemann-hypothesis criterion, the key identity is ζ(s)G(s)=Γ(s), extended from Re(s)>1 to","core_discovery":"Under the assumption that every nontrivial zero of ζ is simple, the paper proves the exact identity Φ(e^{-x}) = Σ_ρ Γ(ρ)x^{-ρ}/ζ'(ρ) + πκ(x) + 2λ(x) + 4υ(x) − 2β(x)log(2πx) − 2. The distinguishing structural feature is that Γ(s)x^{-s}/ζ(s) has double poles at the negative even integers, where poles of Γ collide with trivial zeros of ζ; their residues contain the logarithmic term and the digamma and logarithmic-derivative weights of ζ. The paper also proves an unconditional implication: if Φ(e^{-x})=O(x^{-1/2}) near 0, then its Mellin transform G(s) is holomorphic for Re(s)>1/2, and the identity theorem applied to ζ(s)G(s)−Γ(s) rules out zeros with Re(s)>1/2; the functional equation then rule","pith_inferences":["The ζ(s)G(s)−Γ(s) identity-theorem trick never divides by ζ near a possible zero, so it may convert other decay bounds on Möbius-type averages into zero-free regions for related Dirichlet series.","The paper's closing remark on the e^{-n^α x} family suggests a testable dichotomy: logarithmic residues should appear exactly for rational α and not for irrational α, since Gamma poles and trivial zeta zeros collide only then.","The converse's extra hypotheses indicate the O(x^{-1/2}) condition is far stronger than RH itself, so the criterion is best read as a sufficient condition; it is unlikely to offer an easy path to RH through the converse direction."],"forward_implications":["A proof of the O(x^{-1/2}) bound would immediately settle the Riemann hypothesis, with no separate control of the zeros.","The explicit formula determines the Möbius Laplace transform exactly from the zeros, through Γ(ρ)x^{-ρ}/ζ'(ρ), plus four computable entire functions and a logarithmic term.","The Möbius–Bessel identities give absolutely convergent closed forms for κ, β, and λ as series of J0 with rotated argument, avoiding conditionally convergent sums entirely.","Under RH, simplicity, and the absolute-convergence hypothesis, the criterion is two-way: the x^{-1/2} decay holds if and only if RH holds.","If simplicity is dropped, the formula survives with multiplicity-adjusted derivative residues, so the structural mechanism does not depend on Hypothesis (S)."],"fun_headline_variants":["Möbius decay bound forces all zeros onto critical line","Explicit Möbius transform formula proves RH from decay","Decay of Möbius sum: a new route to the Riemann hypothesis"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the O(x^{-1/2}) bound on the Möbius Laplace sum near 0; it is an unproved regularity condition, and if it fails, the unconditional implication has nothing to act on (while the explicit formula separately leans on the unproved simplicity of all nontrivial zeros).","fun_headline_variants_meta":{"raw":{"variants":["Möbius decay bound forces all zeros onto critical line","Explicit Möbius transform formula proves RH from decay","Decay of Möbius sum: a new route to the Riemann hypothesis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000755,"raw_usage":{"total_tokens":3214,"prompt_tokens":786,"completion_tokens":2428,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2368}},"tokens_in":530,"tokens_out":2428,"duration_ms":14530,"temperature":1.0,"reasoning_tokens":2368,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:55:23.882091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate both sides of the explicit formula at x=0.1, x=0.5, and x=1 to 60 significant digits; any mismatch between the direct Möbius sum and the zero-plus-special-function side would falsify the residue computation. For the RH implication there is no numerical experiment that could directly falsify it—a counterexample would require RH to be false and the O(x^{-1/2}) bound to hold.","supporting_citations":[],"review_version":3}