{"id":"a6106c7a-7446-49b5-8c57-ccdb5e9797ed","arxiv_id":"2512.18936","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For multiplicative functions generated by a fixed sequence of prime-power values, an explicit formula extracts the critical-line contributions of ζ(s)^z ζ(2s)^w and gives a criterion for persistent, apparent, or absent bias.","lead":"This paper studies multiplicative functions whose values on prime powers are prescribed by a single sequence, and derives a new explicit formula for their smoothed sums that separates the main term from oscillatory contributions tied to the zeros of the Riemann zeta function. It then classifies when such sums show persistent bias, apparent bias, or no bias at a natural square-root scale.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The bias classification fails for z=0: a_rho=0 for all rho (sin pi z=0), so the 'apparent bias' and 'unbounded' cases collapse; e.g. epsilon1=0, epsilon2=i gives persistent rather than apparent bias.","rationale":"The reader identified the unproved nonvanishing of a_rho as the weakest assumption. My stress test confirms that this is load-bearing, and shows it is not merely unproved but false in an admissible case: z=0 forces a_rho=0 for every rho. This produces a concrete counterexample to the bias classification: for epsilon1=0, epsilon2=i, the normalized sum converges to a nonzero constant, so the behavior is persistent, while Theorem 5.15(ii) predicts apparent bias. The explicit-formula core of the paper may still be correct, so the right response is to condition acceptance on adding z!=0 (or an equivalent nonvanishing Fourier-coefficient hypothesis) and on treating the integer/zero cases explicitly. Since the reader's verdict was already CONDITIONAL, this does not change the verdict, but it sharpens the required revision.","tokens_in":33161,"tokens_out":18739,"duration_ms":182851,"concrete_test":"Specialize the proof to z=0, w=i by setting epsilon1=0, epsilon2=i, epsilon_k=0 for k>=3. From (311) all a_rho=0, so S(x) in (198) is identically zero and (198) gives B_f^exp(x;0,i)=c_{1/2}(0,i)+o(1). Numerically evaluate B_f^exp(x;0,i)-c_{1/2}(0,i) for x=10^2,...,10^8 using the approximation (203) for G_f, with G_f(1/2) a convergent Euler product of nonzero factors; if the values tend to 0, the pointwise limit exists and Theorem 5.15(ii)'s 'apparent bias' prediction is contradicted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.15 is false as stated because it does not exclude z=0. In (311), a_rho = -sin(pi z)/pi Gamma(1+z) lambda_{rho,0}. If epsilon1=0, then z=0 and every a_rho vanishes, regardless of J_rho(0). The proof's assertion just after Lemma 6.17 that 'a_rho=0 iff J_rho(0)=0, and the latter is impossible' is therefore incorrect for z=0: J_rho(0) is generically nonzero but a_rho=0 because sin(pi z)=0. Consequently the zero-sum term S(x) in (198) is identically zero. Take a finitely supported fake Mobius function with epsilon1=0, epsilon2=i, and epsilon_k=0 for k>=3; then w=i, Re(z+w)=0. Equation (198) becomes B_f^exp(x;0,i)=c_{1/2}(0,i)+o(1), and c_{1/2}(0,i) is nonzero for this choice, so the limit exists and is nonzero. This is persistent bias, not apparent bias as Theorem 5.15(ii) predicts. The same mechanism destroys the unboundedness claim in (iii) for z=0. The theorem needs an explicit hypothesis such as z != 0 plus at least one nonzero Fourier coefficient, or a separate treatment of the integer/zero cases along the lines of Remark 5.16.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'unimodular fake Möbius functions': multiplicative functions f:N→S^1∪{0} whose prime-power values are a prescribed sequence ε_k. Their Dirichlet series factor as F_f(s)=ζ(s)^z ζ(2s)^w G_f(s) with z=ε_1 and w=ε_2-ε_1(ε_1+1)/2, and G_f holomorphic/bounded on Re(s)>1/3. The main analytic claim is an explicit formula for A_f^exp(x)=∑ f(n)e^{-n/x}: subtracting the main contribution from s=1 leaves a secondary term from s=1/2, a sum over nontrivial zeta zeros, and an error. This is derived by contour integration with Hankel contours, followed by Watson-lemma expansions. The paper then defines persistent, apparent, and zero bias at the scale x^{1/2}(Log x)^{w-1} and states a criterion in Theorem 5.15, conditional on RH, SZC, and a weak Mertens-type zero-gap assumption (190).","tokens_in":33472,"tokens_out":9011,"duration_ms":90614,"significance":"The explicit formula, if correct, is a substantive extension of the Selberg-Delange method: it extracts critical-line contributions, is parameter-free in that z,w are read off from the first two coefficients, and reduces to the known integer cases in [19]. The detailed contour computation and the Watson-lemma analysis are valuable and largely coherent. However, the advertised bias classification is not valid as stated: the z=0 case is not excluded, and the proof relies on an unproved nonvanishing assertion for the zero Fourier coefficients. These are load-bearing for Theorem 5.15(ii)-(iii), so the paper needs a corrected and sharpened statement, not merely editorial revision.","major_comments":[{"comment":"The assertion 'a_ρ=0 iff J_ρ(0)=0, and the latter is impossible' is false when z=0. From Eq. (311), a_ρ = -sin(πz)/π Γ(1+z) λ_{ρ,0}, so sin(πz)=0 forces a_ρ=0 for every ρ even if J_ρ(0)≠0. Moreover Eq. (162) gives Δ_ρ(x)≡0 for z=0, so the zero-sum term S(x) in (198) is identically zero. Concretely, take ε_1=0, ε_2=i, ε_k=0 for k≥3; then z=0, w=i, Re(z+w)=0, and (198) yields B_f^exp(x;0,i)=c_{1/2}(0,i)+o(1) with c_{1/2}(0,i)≠0. The pointwise limit therefore exists: this is persistent bias, not apparent bias as Theorem 5.15(ii) claims. The same mechanism invalidates the unboundedness claim in (iii) for z=0. The theorem needs an explicit hypothesis z≠0 (plus at least one nonzero a_ρ), or a separate treatment of the z=0/integer cases as in Remark 5.16.","section":"Theorem 5.15, Eq. (162), Eq. (311), proof after Lemma 6.17"},{"comment":"Even away from z=0, the proof of (ii) and the unboundedness part of (iii) require that some Fourier coefficient a_ρ is nonzero, i.e. that J_ρ(0)≠0 for some zeta zero. The manuscript asserts this is 'impossible (see (163))', but (163) merely defines J_ρ; it gives no nonvanishing proof. J_ρ(0) contains the factor G_f(ρ), and Proposition 3.4 establishes only holomorphy and boundedness of G_f on Re(s)>1/3, not nonvanishing. Since G_f is an Euler product involving g_f(p^{-s}), a zero of g_f in the unit disk could make G_f(ρ)=0 for some ρ; this is not excluded by anything in the paper. The dichotomy in Lemma 6.17 is therefore not established. The bias classification should state an explicit nonvanishing hypothesis on the a_ρ, or prove one.","section":"Eq. (163), Proposition 3.4, Lemma 6.17"},{"comment":"The abstract advertises the explicit criterion as obtained under RH and SZC only, but Theorem 5.15 also assumes the unproved weak-Mertens-type zero-gap condition (190). This assumption is load-bearing: it is used to prove convergence of the zero sum ∑_ρ Δ_ρ(x), the uniform boundedness condition (179), and hence the whole classification. The abstract and the statement of the advertised 'explicit criterion' should include this additional hypothesis, or the theorem should be reorganized so that the unconditional parts and the conditional parts are clearly separated.","section":"Abstract and Theorem 5.15, Assumption 5.11 / Eq. (190)"}],"minor_comments":[{"comment":"Typo: 'if 1≤n=∏_p p^{v_p(n)}, we define we define' should read 'we define'.","section":"Section 3.1"},{"comment":"Typo: 'shaded reigon' should be 'shaded region'.","section":"Figure 1.3.4 caption"},{"comment":"The figure captions are not fully self-contained; for exact reproduction it would help to list the coefficients ε_k and the truncation parameters used in (203).","section":"Section 5 numerical illustrations"}],"recommendation":"major_revision","confidential_remarks":"The explicit-formula part is a genuine contribution and the derivation appears careful. The bias theorem, however, is currently stated too broadly: the z=0 counterexample is decisive, and the nonvanishing step in the proof is a real gap. I would encourage the editor to require the corrected hypotheses and a rewritten Theorem 5.15 before publication. The paper's length and scope also make it advisable to separate the conditional zero-gap input from the main analytic result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real contribution here is the explicit formula (15): for Euler products of the form ζ(s)^z ζ(2s)^w G_f(s), the paper extracts the s=1/2 contribution and the zero-line sum, not just the main term at s=1. That is a substantive extension of the Selberg–Delange method, and the contour derivation is mostly careful. The Watson-lemma treatment of the Laplace integrals is clean, and the reductions to Martin–Mossinghoff–Trudgian in the integer cases are sensible consistency checks. I did not find a hole in the derivation of the explicit formula itself.\n\nThe problem is the bias classification, Theorem 5.15. In the proof of part (ii), the paper claims that a_rho = 0 iff J_rho(0) = 0, and that the latter is impossible. That is wrong. Equation (311) gives a_rho = -sin(πz)/π · Γ(1+z) λ_{rho,0}; when z=0, sin(πz)=0, so a_rho=0 for every zero, regardless of J_rho(0). This is not a harmless edge case: take ε1=0, ε2=i, ε_k=0 for k≥3. Then z=0, w=i, Re(z+w)=0, and c_{1/2}(0,i) is nonzero. The zero sum S(x) is identically zero up to the remainder, so B_f^exp(x;0,i) tends to c_{1/2}(0,i). That is persistent bias, not the apparent bias predicted by (ii). The same mechanism breaks the unboundedness claim in (iii) for z=0.\n\nThe abstract also overstates the result: the 'explicit criterion' depends on the unproved Weak Mertens-type zero-gap condition (190), which is only stated later. And the numerical illustrations are not reproducible from the text — enough detail is missing to redo them.\n\nSo: the explicit formula deserves a serious referee and probably publication; the bias theorem needs a real fix, not just a remark. The fix is straightforward in principle — add a hypothesis excluding z=0 or more generally z∈Z, or treat the integer cases separately — but as stated the theorem is false. I would send this to peer review, and I'd cite the explicit formula with the bias theorem caveated.","headline":"The explicit formula (15) is a genuinely new extension of Selberg–Delange, but Theorem 5.15 is false as stated: at z=0 every a_rho vanishes, so the bias classification collapses and the paper's own example gives persistent bias where it predicts apparent bias.","tokens_in":34031,"tokens_out":4346,"would_cite":true,"duration_ms":40340,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A25","11M06","11M26","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fake Möbius sums obey an explicit formula from zeta zeros, and their bias type is decided by Re(z+w).","keywords":["fake Möbius functions","multiplicative functions","Selberg–Delange method","explicit formula","Riemann zeta function","nontrivial zeros","bias","logarithmic Cesàro mean"],"falsifier":"Compute J_ρ(0)= (ρ-1)^{-z} G_f(ρ) Z_ρ(ρ) ζ(2ρ)^w Γ(ρ) for a chosen f and a few zeta zeros; if G_f(ρ)=0 for even one ρ, the pointwise-limit and unboundedness conclusions in Theorem 5.15 need re-examination. Separately, numerically evaluate B_f^exp(x) for Re(z+w)=0 with c_{1/2}≠0: if it converges to c_{1/2} as x→∞, the apparent-bias classification fails.","tokens_in":32977,"feed_emoji":"🎯","tokens_out":5021,"duration_ms":44578,"temperature":0.7,"pith_summary":"The paper studies multiplicative functions whose prime-power values are arbitrary unit-modulus numbers assigned independently of the prime, and shows their Dirichlet series factor as ζ(s)^z ζ(2s)^w times a holomorphic Euler product. Assuming RH, simple zeta zeros, and a weak Mertens-type gap condition, it derives an explicit formula for the smoothed summatory function: a main term from s=1, a secondary term from s=1/2, an oscillatory sum over nontrivial zeros, and a small error. From this formula it derives an explicit bias criterion at scale x^{1/2}(Log x)^{w-1}: persistent bias if Re(z+w)>0, apparent bias if Re(z+w)=0, and unboundedness if Re(z+w)<0, with the constant c_{1/2}(z,w) determining the value. This extends the Selberg–Delange method to capture contributions from the critical line, and provides a tunable family of examples for prime-race-type phenomena.","feed_headline":"One sign rule sorts fake Möbius bias","feed_subtitle":"Persistent, apparent, or no bias at the square-root scale depends on Re(z+w), via a new explicit formula.","key_machinery":"The central object is the zeta factorization F_f(s)=ζ(s)^z ζ(2s)^w G_f(s) with z=ε_1, w=ε_2-ε_1(ε_1+1)/2, together with the explicit formula that extracts, in addition to the s=1 main term, the secondary term from s=1/2 and a Laplace-integral term from each nontrivial zero ρ. Watson's lemma converts each such term into a power-of-log expansion, and the zero sum defines an almost-periodic function whose Fourier coefficients encode the bias.","core_discovery":"For a multiplicative f with f(p^k)=ε_k, writing z=ε_1 and w=ε_2-ε_1(ε_1+1)/2, the paper proves (under RH, SZC, and a convergence condition) that A_f^exp(x) - Δ_1(x) = Δ_{1/2}(x) + Σ_ρ Δ_ρ(x) + O(x^a), where Δ_1, Δ_{1/2}, Δ_ρ are explicit Laplace integrals with Watson-type asymptotics. Consequently the normalized difference has limit c_{1/2}(z,w) when Re(z+w)>0, bounded logarithmic-Cesàro oscillation to c_{1/2} when Re(z+w)=0, and unbounded growth when Re(z+w)<0.","pith_inferences":["A natural testable extension is to compute c_{1/2}(z,w) for periodic ε_k sequences and check whether it vanishes on a positive-measure surface in (z,w); the paper leaves the structure of its zeros unexplored.","The argument that the almost-periodic zero sum cannot converge to a limit unless all its Fourier coefficients vanish suggests the no-bias regime requires c_{1/2}(z,w)=0, but also presupposes at least one nonzero coefficient; a concrete check is evaluating G_f(ρ) for a zeta zero under a simple ε_k sequence.","The method likely transfers to Dirichlet L-functions or Dedekind zeta functions, where the analogue of the factorization would involve powers of L(s,χ), giving a new family of bias results for prime races."],"forward_implications":["For completely multiplicative functions f(n)=ξ^{Ω(n)}, the bias type is explicitly determined by ξ alone, since (z,w) are polynomial in ξ; this yields an infinite family with persistent, apparent, or unbounded behavior at the square-root scale.","The expansion extracts contributions from the critical line Re(s)=1/2 that the classical Selberg–Delange method does not; the same machinery applies to any Dirichlet series of the form ζ(s)^z ζ(2s)^w G(s).","The bias criterion reduces to the known rules for ε_k ∈ {0,±1}, so earlier results for classical fake Möbius functions appear as special cases.","Higher coefficients ε_k (k≥3) enter only at a strictly lower order, so the two leading prime-power values determine the bias type."],"fun_headline_variants":["Re(z+w) decides fake Möbius bias","Bias of fake Möbius: sign of Re(z+w)","Fake Möbius bias: persistent or not by Re(z+w)","Explicit formula unifies fake Möbius bias","New rule: Re(z+w) sorts fake Möbius bias"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The classification collapses if every Fourier coefficient a_ρ in the zero sum vanishes (equivalently, if G_f(ρ)=0 at every zeta zero for some admissible sequence), which the paper asserts is impossible without proof; the zero sum also requires an unproved lower bound on gaps between zeta zeros to converge.","fun_headline_variants_meta":{"raw":{"variants":["Re(z+w) decides fake Möbius bias","Bias of fake Möbius: sign of Re(z+w)","Fake Möbius bias: persistent or not by Re(z+w)","Explicit formula unifies fake Möbius bias","New rule: Re(z+w) sorts fake Möbius bias"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000989,"raw_usage":{"total_tokens":4040,"prompt_tokens":764,"completion_tokens":3276,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":3197}},"tokens_in":508,"tokens_out":3276,"duration_ms":22963,"temperature":1.0,"reasoning_tokens":3197,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T14:51:09.780194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute J_ρ(0)= (ρ-1)^{-z} G_f(ρ) Z_ρ(ρ) ζ(2ρ)^w Γ(ρ) for a chosen f and a few zeta zeros; if G_f(ρ)=0 for even one ρ, the pointwise-limit and unboundedness conclusions in Theorem 5.15 need re-examination. Separately, numerically evaluate B_f^exp(x) for Re(z+w)=0 with c_{1/2}≠0: if it converges to c_{1/2} as x→∞, the apparent-bias classification fails.","supporting_citations":[],"review_version":1}