{"id":"e4d85312-fae4-4681-a66d-1c2f6b3e24cf","arxiv_id":"2608.12257","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For every epsilon>0 and fixed pi', L(s,pi) and L(s,pi×pi') are claimed to be lower bounded and zero-free in a c C_pi^{-epsilon} neighborhood of Re(s)=1, with ineffective c.","lead":"Automorphic L-functions are claimed to admit a new Siegel-type zero-free region near Re(s)=1 whose width decays like any small power of the conductor. If correct, this would be the first uniform result of this strength for all standard L-functions, improving prime number theorems and Brauer-Siegel type asymptotics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The false identity (4.7) L(s,eπ×eπ')=L(s,π×π') is load-bearing; it underpins Prop. 4.6 and Prop. 5.1. The reader's stated identity in Prop. 5.1 is actually correct; the real flaw is (4.7).","rationale":"The stress-test pass found the reader's primary objection—the identity in Proposition 5.1 with |λ_{eπ}+λ_{π'}|^2—to be a miscomputation: the identity is actually correct, so the reader's stated weakest assumption does not land. However, a different and genuine error, identity (4.7), is load-bearing: it is false for non-self-dual representations and is used essentially in Proposition 4.6 (the Landau–Page uniqueness step that reduces the proof to the two-zero configuration (6.3)) and in Proposition 5.1 (the estimate (5.2) providing the factor 1−β_1). In Proposition 4.6, (4.7) makes zeros of L(s,π_1×π_3)L(s,π_2×π_3) appear also as zeros of L(s,eπ_1×eπ_3)L(s,eπ_2×eπ_3), forcing four real zeros of L(s,Π×tilde Π) and contradicting Lemma 4.5; without it, there is no contradiction. In Proposition 5.1, (4.7) is used to assert that β_1 is the greatest real zero of L(s,eπ×eπ') and that 1−β_1 is a double zero of L(s,Π×tilde Π); if L(s,eπ×eπ') has distinct real zeros near 1, the explicit formula bound (5.2) can fail. Both propositions are essential to the proof of Theorem 1.2, so the central claim is not established as written. The proposed concrete test, checking (4.7) for a quartic Dirichlet character, immediately falsifies (4.7), confirming the concern. The verdict should remain REJECT, consistent with the reader's overall judgment, though for a different reason than the reader's primary stated objection.","tokens_in":24317,"tokens_out":25174,"duration_ms":178587,"concrete_test":"Verify (4.7) on GL(1): let π=χ be a primitive Dirichlet character of order 4 (so χ(p) can equal i), and let π' be the trivial character. Then L(s,π×π')=L(s,χ) has Euler factor (1−χ(p)p^{-s})^{-1} at unramified p, while L(s,eπ×eπ')=L(s,\\bar χ) has Euler factor (1−\\bar χ(p)p^{-s})^{-1}. Choose p with χ(p)=i; the factors are (1−i p^{-s})^{-1} and (1+i p^{-s})^{-1}, which differ. This directly disproves (4.7).","verdict_should_be":"REJECT","load_bearing_attack":"The reader's weakest assumption—that Proposition 5.1 uses a false local identity for |λ_{eπ}(p)+λ_{π'}(p)|^2—is a miscomputation. For Π=eπ⊞π', tilde Π=π⊞eπ', the four Euler factors give Λ_{Π×tilde Π}(p)/log Np = λ_{eπ}(p)λ_π(p)+λ_{eπ}(p)λ_{eπ'}(p)+λ_{π'}(p)λ_π(p)+λ_{π'}(p)λ_{eπ'}(p) = |λ_π(p)|^2+|λ_{π'}(p)|^2+2Re(λ_π(p)λ_{π'}(p)) = |λ_{eπ}(p)+λ_{π'}(p)|^2 (since λ_{eπ}(p)=\\overline{λ_π(p)}). So that identity is correct. The actual load-bearing flaw is (4.7): L(s,eπ×eπ')=L(s,π×π'), which is false for non-real L-functions; e.g., for GL(1) characters χ,ψ, L(s,\\bar χ\\bar ψ)≠L(s,χψ). Proposition 4.6 invokes (4.7) to turn zeros of L(s,π_1×π_3)L(s,π_2×π_3) into zeros of L(s,eπ_1×eπ_3)L(s,eπ_2×eπ_3), obtaining four real zeros of L(s,Π×tilde Π) and a contradiction with Lemma 4.5. Without (4.7), the zeros need not coincide, so the reduction to the configuration (6.3) with β_ε<β_1 fails. Proposition 5.1 also uses (4.7) to identify β_1 as the greatest real zero of L(s,eπ×eπ') and to make 1−β_1 a double zero of L(s,Π×tilde Π); both steps are unsupported. Because Propositions 4.6 and 5.1 are essential to Theorem 1.2, the central claim is not established as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to establish, for every ε > 0, ineffective constants c and c' such that the standard L-function L(s,π) and the Rankin–Selberg L-function L(s,π×π') (with π' fixed) satisfy lower bounds of size (C_π(|t|+3))^{-ε} in a one-sided region σ ≥ 1 - c(C_π(|t|+3))^{-ε}. The method combines two Turán-type power-sum inequalities (Propositions 2.1 and 2.2) with an auxiliary isobaric representation, a Landau–Page-type zero-separation result (Proposition 4.6), and a zero-repulsion estimate (Proposition 5.1) that are intended to force the crucial factor (1−β1). Applications to prime number theorems and Brauer–Siegel-type statements are also presented.","tokens_in":24785,"tokens_out":18253,"duration_ms":145199,"significance":"If the main theorems were correct, they would constitute a major advance: the first unconditional Siegel-type zero-free regions with conductor-power saving for all GL(n) standard and Rankin–Selberg L-functions, subsuming prior results of Brumley and of Harcos–Thorner. The two-pronged power-sum strategy is original, and the paper is clearly organized with careful use of standard analytic number theory tools. However, the proof depends on at least two false L-function identities and one unjustified estimate, so the central claims are not established as written.","major_comments":[{"comment":"The identity L(s,eπ×eπ') = L(s,π×π') is false in general. The correct relation is L(s,eπ×eπ') = \\overline{L(\\bar{s},π×π')}, and for GL(1) Hecke characters χ,ψ the asserted identity would say L(s,\\bar{χ}\\bar{ψ}) = L(s,χψ), which fails whenever χψ is non-real. This identity is used in the proof of Proposition 4.6 to convert zeros of L(s,π1×π3)L(s,π2×π3) into zeros of L(s,eπ1×eπ3)L(s,eπ2×eπ3), and in the proof of Proposition 5.1 to identify β1 as a double zero of L(s,Π×eΠ). Without (4.7), the four-zero contradiction with Lemma 4.5 and the asserted (1−β1) factor are unsupported.","section":"§4.2, Eq. (4.7)"},{"comment":"The proof asserts that |λ_{eπ}(p)+λ_{π'}(p)|^2 log Np = Λ_{Π×eΠ}(p) by (4.11), but the four Euler factors actually give Λ_{Π×eΠ}(p)/log Np = |λ_π(p)|^2 + |λ_{π'}(p)|^2 + 2Re(λ_π(p)λ_{π'}(p)), whereas the left-hand side equals |λ_π(p)|^2 + |λ_{π'}(p)|^2 + 2Re(\\overline{λ_π(p)}λ_{π'}(p)). These expressions differ whenever Im(λ_π(p))Im(λ_{π'}(p)) ≠ 0, so the inequality used to bound the desired sum by S(x;Π) does not follow from (4.11). This is a second load-bearing error in the proof of (5.2).","section":"§5, proof of Proposition 5.1"},{"comment":"The estimate (x^{−12000β1}−x^{−β1})Γ(−β1) ≪ (1−β1)^{−1}x^{−β1} ≪ 1−β1 is not justified. For β1 close to 1, Γ(−β1) ≍ (1−β1)^{−1}, and the hypotheses x ≥ (CπCπ')^{c25} and (5.1) do not imply x^{−β1} ≤ C(1−β1)^2. If 1−β1 is exceptionally small, this term can be much larger than the claimed O((1−β1)(log x)^3), so the bound (5.2) is not established even if the earlier identity issues were repaired.","section":"§5, proof of Proposition 5.1, contribution of 1−β1"}],"minor_comments":[{"comment":"The sentence containing 'Otherwise,C pi is bounded' contains a typo: it should read 'C_π'.","section":"§6.1, Eq. (6.1)"},{"comment":"The quantity N_ε = exp(K/(300ε)) is used before it is formally introduced in Proposition 6.3; flagging it in the strategy section is helpful, but the forward reference is not a substantive issue.","section":"§2.2, Step 2"}],"recommendation":"reject","confidential_remarks":"I would not recommend a major revision because the central mechanism relies on an identity that is false for the very non-self-dual cases the theorem intends to cover, and replacing it by the correct functional-equation relation changes the zero structure so fundamentally that Propositions 4.6 and 5.1 would both need qualitatively new inputs. The flaws are too deep for local repairs within the scope of this manuscript, despite the interesting overall strategy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorems are a big deal if they hold: uniform Siegel-type zero-free regions for all GL(n) standard L-functions and Rankin-Selberg L-functions with one factor fixed, with conductor-strength epsilon rather than the much worse Brumley exponent. The power-sum mechanism, pairing Turan's second main theorem with the Sos-Turan real-part lower bound, is genuinely new and clearly explained in Section 2. The applications to prime number theorems and Brauer-Siegel type results are natural and, conditional on the zero-free region, straightforward.\n\nThe reader's specific objection to Proposition 5.1 is a miscomputation. For Pi = e-pi ⊞ pi', tilde-Pi = pi ⊞ e-pi', the four Euler factors do give Lambda_{Pi×tilde-Pi}(p) = (lambda_{e-pi}(p)+lambda_{pi'}(p))(lambda_pi(p)+lambda_{e-pi'}(p)) log Np = |lambda_{e-pi}(p)+lambda_{pi'}(p)|^2 log Np. That identity is fine.\n\nThe real problem is (4.7): L(s,e-pi×e-pi') = L(s,pi×pi') is false for non-real L-functions. For real s, L(s,e-pi×e-pi') is the complex conjugate of L(s,pi×pi'), not equal to it. The paper uses (4.7) in Proposition 4.6 and Proposition 5.1 to move real zeros between L(s,pi×pi') and L(s,e-pi×e-pi') and to make 1-beta_1 a double zero. Those specific inferences are actually valid, because they only concern real zeros and the conjugation property gives the same conclusion on the real axis. So the error is real but appears repairable: replace (4.7) by the conjugation identity and verify every invocation involves only real s. The written proof is not correct as it stands, but the strategy does not seem to collapse.\n\nThe zero-repulsion parts, especially the residue computation in Proposition 5.1 and the use of Moreno's Lemma 4.12, are delicate and deserve careful referee scrutiny. I did not find a second concrete error, but the 1-beta_1 factors are handled with several nontrivial estimates. The ineffectivity is openly flagged, and the citation pattern looks appropriate.\n\nThis paper is for analytic number theorists working on L-functions and zero-free regions. It deserves serious peer review, not desk rejection. A good referee should require a corrected treatment of (4.7) and a close check of Section 5. If the repair works, this is a major advance.","headline":"The power-sum mechanism is genuinely novel and the claimed theorems would be major, but the paper contains a false L-function identity (4.7) that is used in central places, though it looks repairable because only real zeros are involved.","tokens_in":25342,"tokens_out":7678,"would_cite":false,"duration_ms":62039,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11F66","11M26","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes uniform conductor-power Siegel-type zero-free regions for all standard and Rankin–Selberg L-functions, without assuming modularity of tensor products.","keywords":["zero-free regions","Rankin-Selberg L-functions","power sums","Siegel zeros","automorphic representations","Brauer-Siegel theorem","prime number theorems","Turán theory"],"falsifier":"Because the constant c_25 in Proposition 5.1 is effectively computable, the inequality (5.2) can be tested numerically for a concrete pair (π,π′) with explicitly known Hecke eigenvalues, such as F=Q, π′=1, and π the L-function of an elliptic curve with rational coefficients. Compute both sides at x=(C_π C_{π′})^{c_25} using a rigorous zero-finder to locate the greatest real zero β_1; a violation of (5.2) would identify a defect in the proof, while repeated agreement across several conductors would corroborate the mechanism.","tokens_in":24072,"feed_emoji":"🔢","tokens_out":22243,"duration_ms":219792,"temperature":0.7,"pith_summary":"This paper develops a new method for zero-free regions near Re(s)=1 for standard and Rankin–Selberg L-functions attached to cuspidal automorphic representations. The central claim is that for every fixed π′ and every ε>0, there exists an ineffective constant c such that L(s,π×π′) has no zeros and satisfies |L(σ,π×π′)| ≥ c $C_π^{{-ε}}$ in the region σ ≥ 1 − c $C_π^{{-ε}}$; the case π′=1 gives the same for standard L-functions. The method replaces the classical nonnegative-coefficient auxiliary series, whose utility is limited by unproved modularity of Rankin–Selberg products, with two complementary power-sum lower bounds. If correct, these are the first unconditional conductor-power Siegel-type zero-free regions for all GL(n) L-functions, and they imply improved prime number theorems and new Brauer–Siegel-type results.","feed_headline":"Power sums yield uniform Siegel-type zero-free regions","feed_subtitle":"A two-part power-sum argument removes the modularity barrier to conductor-power zero-free regions for these L-functions.","key_machinery":"The machinery is the pairing of two power-sum inequalities from Section 2. The first, Proposition 2.1, asserts that for any complex numbers z_1,...,z_ν there exists a power ℓ ∈ [K,2K] with |z_1^ℓ+···+z_ν^ℓ| ≥ (|z_1|/50)^ℓ; it is used to extract a single exceptional zero β_ε from the logarithmic derivative G_k of F(z)=L(z,π_ε×π′)L(z,π_ε×~π). The second, Proposition 2.2, is a real-part power-sum bound with nonnegative weights that gives an index j with Re(Σ b_n z_n^j) ≥ b_1|z_1|^j/8; via the zero-repulsion lemma it supplies the factor 1−β_1 that bounds the same G_k from above. Matching the two bounds forces 1−β_1 to be at least a fixed power of $C_π^{{-1}}$, closing the zero-free region.","core_discovery":"The discovery, on the paper's own terms, is that two lower bounds for power sums can serve as the sole engine for zero-free regions: the first power-sum inequality detects the presence of an exceptional zero through the logarithmic derivative of a shifted Rankin–Selberg product, while a real-part power-sum bound, in combination with a zero-repulsion lemma, controls how close any other zero can come to that exceptional zero. The interaction of these two bounds forces the exceptional zero to lie at distance at least a fixed power of the conductor from 1, producing the bound |L(σ,π×π′)| ≥ c $C_π^{{-ε}}$ throughout σ ≥ 1 − c $C_π^{{-ε}}$. Standard analytic properties of Rankin–Selberg L-functions are used only as input, and the classical Siegel–Tatuzawa bound for Dirichlet L-functions is recovered as a special case.","pith_inferences":["If the power-sum mechanism is as robust as the paper suggests, any future improvement in the constants of the underlying power-sum inequalities would transfer directly to wider zero-free regions, effectively decoupling progress on Siegel zeros from progress on functoriality.","The same two-inequality architecture might apply to L-functions without a full Rankin–Selberg theory, such as symmetric powers, provided a zero-repulsion statement can be established for them; the paper does not claim this extension.","Because the region width is C_π^{-ε}, the proof is uniform in the eigenvalue aspect as well as the conductor aspect; one could test whether the argument adapts to the spectral aspect for Maass forms, where the conductor grows differently.","The paper states, with proof deferred to forthcoming work, that nonnegativity-based methods cannot reach the main theorem without an unproven modularity hypothesis; if that meta-claim is correct, the power-sum route is not merely an alternative but the only currently viable path to such unconditional results."],"forward_implications":["For self-dual π, the prime number theorem error E(x;π) satisfies E(x;π) ≤ c e^{-√log x} as soon as log x ≥ c C_π^ε, matching the quality of classical results for real Dirichlet characters.","For Rankin–Selberg prime sums, new ‘highly uniform’ prime number theorems hold, with error term either e^{-√log x} or (log x)^{-1/ε} depending on the self-duality of the factors.","The theorem yields Brauer–Siegel-type limits: along any sequence with C_{π_j} → ∞, we have log|L(1,π_j)| / log C_{π_j} → 0 and the analogous statement for L(1,π_j×π′).","The result subsumes the twist-aspect Siegel-type bounds and improves the previous conductor-exponent in the fixed-π′ aspect from a fixed power of C_π to an arbitrary ε-power.","In the conjectural Bloch–Kato framework for weight −2 motives, the lower bound gives polynomial-in-conductor control of the Shafarevich–Tate group order, paralleling the classical consequence for class numbers and regulators."],"supporting_citations":[{"why":"Supplies the two power-sum propositions that drive zero detection and zero repulsion.","marker":"[29]"},{"why":"Provides the zero-repulsion lemma that yields the crucial factor of 1−β_1.","marker":"[30]"},{"why":"Gives the nonnegativity and Landau-type lemmas for isobaric sums, and frames the modularity hypothesis that the new method avoids.","marker":"[15]"},{"why":"Supplies the narrow, effectively computable zero-free regions used as base cases and in contour shifts.","marker":"[5]"},{"why":"Together with [13], supplies the twist-aspect Siegel-type bound used in the initial reductions.","marker":"[12]"},{"why":"Provides the twist-aspect zero-free region for Rankin–Selberg L-functions, used with [12] to eliminate non-entire cases.","marker":"[13]"},{"why":"Defines the Rankin–Selberg L-functions and their local factors used throughout the proof.","marker":"[20]"},{"why":"Gives the starting nonvanishing of Rankin–Selberg L-functions on the line Re(s)=1.","marker":"[36]"},{"why":"Supplies the bound on high prime-power coefficients used in the upper bound for the logarithmic-derivative expression.","marker":"[24]"}],"fun_headline_variants":["Power sums alone produce zero-free regions","Conductor-power zero-free regions via power sums","Power-sum bounds unlock new zero-free regions","Zero-free regions without modularity assumptions","Power sums drive Siegel-type zero-free regions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on the zero-repulsion estimate for the isobaric object Π = eπ ⊞ π′: every zero of L(s,Π×eΠ) other than the exceptional one is at distance at least a constant multiple of (1−β_1)/log(conductor) from s=1, uniformly in the conductors.","fun_headline_variants_meta":{"raw":{"variants":["Power sums alone produce zero-free regions","Conductor-power zero-free regions via power sums","Power-sum bounds unlock new zero-free regions","Zero-free regions without modularity assumptions","Power sums drive Siegel-type zero-free regions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000145,"raw_usage":{"total_tokens":1198,"prompt_tokens":984,"completion_tokens":214,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":149}},"tokens_in":600,"tokens_out":214,"duration_ms":2422,"temperature":1.0,"reasoning_tokens":149,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:14:37.878183+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Because the constant c_25 in Proposition 5.1 is effectively computable, the inequality (5.2) can be tested numerically for a concrete pair (π,π′) with explicitly known Hecke eigenvalues, such as F=Q, π′=1, and π the L-function of an elliptic curve with rational coefficients. Compute both sides at x=(C_π C_{π′})^{c_25} using a rigorous zero-finder to locate the greatest real zero β_1; a violation of (5.2) would identify a defect in the proof, while repeated agreement across several conductors would corroborate the mechanism.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two power-sum propositions that drive zero detection and zero repulsion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the zero-repulsion lemma that yields the crucial factor of 1−β_1."},{"cited_title":"Hoffstein and D","cited_arxiv_id":null,"evidence_quote":"Gives the nonnegativity and Landau-type lemmas for isobaric sums, and frames the modularity hypothesis that the new method avoids."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the narrow, effectively computable zero-free regions used as base cases and in contour shifts."},{"cited_title":"Harcos and J","cited_arxiv_id":null,"evidence_quote":"Together with [13], supplies the twist-aspect Siegel-type bound used in the initial reductions."},{"cited_title":"Harcos and J","cited_arxiv_id":null,"evidence_quote":"Provides the twist-aspect zero-free region for Rankin–Selberg L-functions, used with [12] to eliminate non-entire cases."},{"cited_title":"Jacquet, I","cited_arxiv_id":null,"evidence_quote":"Defines the Rankin–Selberg L-functions and their local factors used throughout the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the starting nonvanishing of Rankin–Selberg L-functions on the line Re(s)=1."}],"review_version":1}