{"id":"8acb4c8a-21ef-4cc5-a3de-69f9296dc438","arxiv_id":"2412.02416","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A weighted zero-density estimate in the spectral aspect for GL(3) Hecke-Maass L-functions is derived from a twisted second moment asymptotic.","lead":"The paper proves a new bound on how many GL(3) automorphic L-functions have zeros lying unusually far to the right of the critical line, averaged over a spectral family. If correct, it gives number theorists a tool for replacing unproven Riemann Hypothesis assumptions in certain GL(3) problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's proof conflates the box height H in (1.3) with the spectral normalizer H_spec in (1.8); substituting Proposition 1.4 consistently leaves an unsuppressed factor H_spec, so the stated bound does not follow.","rationale":"The reader's weakest assumption—the H versus H_spec normalization conflict—is exactly the point I would identify as load-bearing. It appears in the final step of the proof of Theorem 1.1 and is not repaired anywhere afterward. The paper contains substantial positive work: the Kuznetsov-based twisted second moment computation has the expected diagonal terms, and the error-term analysis follows established GL(3) techniques. But the central theorem depends on a notational identification that the text does not make and cannot make, since H_spec is fixed of size T^3M^2 while the box height H ranges down to 3/logT. I also credit the reader's secondary concern about the horizontal edge: for H ≈ 3/logT the sinh weight in Lemma 4.1 is exponentially large in logT, and noting that the vertical side inside the critical strip involves only log|ω| does not control the top edge. Both issues concern the central claim rather than peripheral lemmas, so I see no reason to alter the reader's rejection. If the normalization is corrected by redefining N(σ,H), the stated uniformity range and the size of the bound would have to change; if it is not corrected, the theorem is unsupported.","tokens_in":31127,"tokens_out":10639,"duration_ms":118075,"concrete_test":"Re-derive the closing display of §7 with two distinct symbols, h_box for the height in (1.3) and h_spec for the normalizer in (1.8). Substitute (1/h_spec)∑|LM|² ≤ 1+O(T^{-2δσ}) from Proposition 1.4, and keep the prefactor 1/h_box coming from N(σ,H). Check whether the resulting contribution h_spec can be bounded by h_box T^{-δσ} for 3/logT < h_box ≤ T^{θ1}; since h_spec ≍ T^{3+2θ}, this fails whenever 3+2θ > θ1. The same two-symbol computation should be applied to the horizontal-edge integral with h_box ≈ 3/logT to test whether the exponential sinh weight is compensated by the second-moment bound; if not, the small-H uniformity claim also fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem is gated by a normalization mismatch. In Theorem 1.1, H is the box height in N(σ,H) (Eq. 1.3), with 3/logT < H < T^{θ1}. In Proposition 1.4, H is redefined by (1.8) as H_spec := (1/192π^5)∫ h_{T,M}(µ)d_specµ, of size ≍ T^3M^2. The final step of §7 replaces (1/H)∑_j (h_{T,M}/N_j)|LM(·)|² by the Proposition's 1+O(T^{-2δσ}). That replacement is valid only with denominator H_spec. Keeping the two symbols distinct, Proposition 1.4 gives (1/H_spec)∑|LM|² ≤ 1+O(T^{-2δσ}), hence (1/H_box)∑|LM|² ≤ (H_spec/H_box)(1+O(T^{-2δσ})). After the t-integral of length H_box, this term is O(H_spec), not O(H_box). Since H_spec ≍ T^3M^2 = T^{3+2θ}, the displayed chain cannot yield N(σ,H) ≪ H T^{-δσ}logT for H ≤ T^{θ1} with θ1 small. No displayed cancellation from Corollary 7.2 removes this factor. A secondary gap is the horizontal-edge integral for H ≈ 3/logT: the sinh weight in Lemma 4.1 is of size exp(π(1-1/logT)/(2H)) ≈ T^{π/6}, and Remark 7's observation that one vertical side inside the critical strip uses only Re log does not control this top edge.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Kuznetsov-based approach to the spectral second moment of GL(3) Hecke–Maass L-functions. It states an asymptotic formula (Theorem 1.2) for the twisted second moment with the localizing weight h_{T,M}, derives from it a mollified second moment bound (Proposition 1.4), and then uses a Selberg/Conrey–Soundararajan argument-principle lemma to claim a weighted zero-density estimate (Theorem 1.1) of the form N(σ,H) ≪ H T^{-δσ} log T, uniformly in 3/log T < H < T^{θ1}. The proof follows the standard route: approximate functional equations, the GL(3) Kuznetsov trace formula, character sum estimates, Eisenstein contribution bounds via subconvexity, and contour shifts for the mollified moments.","tokens_in":31297,"tokens_out":12237,"duration_ms":139019,"significance":"If Theorem 1.1 were valid, it would be a first spectral-aspect zero-density estimate of Selberg type for GL(3) automorphic L-functions, and the twisted/mollified second moment calculations in Theorems 1.2 and Proposition 1.4 would be valuable technical ingredients. The paper is largely parameter-free: the main terms come from residues, the test function is explicit, and I saw no circular use of the target result or fitted constants. The arithmetic sections appear carefully executed and cite the relevant subconvexity and spectral tools. However, the central deduction in Section 7 contains a normalization mismatch between the box height in (1.3) and the spectral normalizer in (1.8), and a separate horizontal-edge issue affects the small-H range. As a result, the advertised theorem does not follow from the established estimates. The component results may still be of interest, but the paper's central claim is unsupported.","major_comments":[{"comment":"The proof of Theorem 1.1 conflates the box height H in (1.3) and Lemma 4.1 with the spectral normalizer H defined in (1.8). Proposition 1.4 establishes a bound for (1/H_spec)Σ_j (h_{T,M}/N_j)|LM(1/2+σ+iτ,φ_j)|^2, where H_spec = (1/192π^5)∫ h_{T,M} dμ_spec ≍ T^3 M^2. When this bound is substituted into the displayed estimate in Section 7, the factor 1/H_spec does not cancel with the box height H_box. Keeping the two symbols distinct gives (1/H_box)Σ_j(h/N_j)|LM(·)|^2 ≤ (H_spec/H_box)(1+O(T^{-2σδ})). The t-integral of length H_box then leaves a term of order H_spec log T = O(T^{3+2θ} log T), and no displayed cancellation from Corollary 7.2 removes this factor. Since Theorem 1.1 requires an upper bound O(H T^{-δσ} log T) with H ≤ T^{θ1}, the claimed theorem does not follow from the given estimates. If H in (1.3) is instead intended to be the spectral normalizer, then the box height in N(σ,H;φ_j) is left undefined and the uniformity range in H loses its meaning.","section":"Section 7; Eqs. (1.3), (1.7)–(1.8)"},{"comment":"The horizontal-edge contribution is not controlled in the range H ≈ 3/log T. In Lemma 4.1, the second integral over α has integrand sinh(π(α-W0)/(2H)) log|ω(α± iH)|. In the application, W0 = 1/log T and α runs to 1, so at α = 1 the sinh factor is of size exp(π(1-1/log T)/(2H)) ≍ T^{π/6}. The proof replaces log|x| by (|x|^2-1)/2, so this term requires a bound on the spectral average of |LM(1/2+α+iH)|^2 that would offset this large weight. Proposition 1.4 gives no such decay. Remark 7 asserts that only the real part of the logarithm appears in the part of the integral inside the critical strip, but that observation concerns the vertical side and does not control the horizontal sides. Thus the claimed uniformity in H down to 3/log T is not supported by the argument in the text.","section":"Section 7; Lemma 4.1 and Remark 7"}],"minor_comments":[{"comment":"The statement says “Let 2/log T < σ < 1/2” but the uniformity range is written as 3/log T < H < T^{θ1}; the relation between these thresholds and the choice W0 = 1/log T in Section 7 should be clarified.","section":"Theorem 1.1, Eq. (1.3)"},{"comment":"The sentence “box bounded by 1/2+1/log T ± 2iH and 3/2 ± 2iH” is inconsistent with the later choice W0 = 1/log T and W1 = 1; a change of variables or a precise definition of the box coordinates should be stated explicitly.","section":"Section 7, first paragraph"},{"comment":"The phrase “provided for δ ≤ 1/3” is garbled; it should presumably read “provided δ ≤ 1/3”.","section":"Section 6, Eq. (6.23)"},{"comment":"The abstract advertises applications of the zero-density estimate, but Remark 2 defers the proof of the application to the second author's thesis. This makes the claimed relevance non-verifiable in the present paper.","section":"Remark 2"}],"recommendation":"reject","confidential_remarks":"The referee report reflects the main technical issue: the zero-density theorem is not derived because of the normalization mismatch between the box height and the spectral normalizer. This is not a typographical issue but a structural one, and the horizontal-edge problem is separate. The twisted and mollified second moment computations may have independent value, but the paper's advertised main result is not established. A future revision would need to reformulate the theorem and its proof architecture substantially, so the appropriate recommendation for the present submission is reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sun and Wang have real results buried here, but the headline theorem is not one of them as stated. The twisted second moment asymptotic (Theorem 1.2) and the mollified second moment bound (Proposition 1.4) are serious work: the GL(3) Kuznetsov machine is deployed carefully, the arithmetic sums are handled with the expected technology, and the citations are appropriate. This is also, as far as I know, the first attempt at a spectral-aspect zero-density estimate for GL(3) Maass forms, and the analogy with the Liu–Streipel GL(2) result is well drawn.\n\nThe problem is that Theorem 1.1 does not follow from the proof. In (1.3), H is the box height, with 3/log T < H < T^{theta1}. In Proposition 1.4, H is redefined as the spectral normalizer H_spec = (1/192pi^5) integral h_{T,M} d_spec mu, which is of size about T^3 M^2. Section 7 substitutes the proposition into the argument-principle estimate using the same symbol and concludes N(sigma,H) << H T^{-delta sigma} log T. Keeping the two symbols separate, Proposition 1.4 bounds (1/H_spec) sum |LM|^2, not (1/H_box) sum |LM|^2. The displayed chain in Section 7 silently loses the factor H_spec/H_box. Corollary 7.2 does not cancel it, and no other displayed estimate removes it. The stated bound cannot be obtained this way.\n\nThere is also a secondary gap at the lower end of the H-range. For H near 3/log T, the sinh weight on the horizontal edge in Lemma 4.1 is about T^{pi/6} in size, and the second-moment bound contributes a main term of size 1, which does not cancel that weight. Remark 7 points to the real-part observation, but that observation alone does not control the top edge. The small-H end needs either a separate argument or a different theorem statement.\n\nI am not saying the idea is hopeless. The twisted and mollified second moments may well be correct, and a corrected normalization or a revised zero-density statement might salvage the application. But the central theorem as stated is not proven. The paper deserves a serious referee: there is too much working machinery here to desk-reject, but the referee should be asked to check the normalization very carefully and to obtain a genuine estimate for the horizontal edge before acceptance.","headline":"A technically serious paper whose stated main theorem is not proven as written: Section 7 conflates the box height H with the spectral normalizer H_spec, and the small-H end of the uniform range is not controlled.","tokens_in":32051,"tokens_out":11460,"would_cite":false,"duration_ms":123663,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F12","11F66","11F67","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"On a spectrally localized family of GL(3) Hecke–Maass cusp forms, almost no L-functions have zeros to the right of $1/2+\\sigma$ up to height $H$, for $H$ as small as $3/\\log T$.","keywords":["GL(3) L-functions","Hecke–Maass cusp forms","zero-density estimate","spectral aspect","Kuznetsov trace formula","twisted second moment","mollification","argument principle"],"falsifier":"Keep every normalization explicit when Proposition 1.4 is substituted into the argument-principle inequality in Section 7; if the unsuppressed spectral factor $H_{\\mathrm{spec}}\\asymp T^3M^2$ remains after multiplying by the box height $H$, the stated bound $N(\\sigma,H)\\ll H T^{-\\delta\\sigma}\\log T$ cannot follow, and a corrected normalization or a different right-hand side would be needed.","tokens_in":30720,"feed_emoji":"📉","tokens_out":19868,"duration_ms":156183,"temperature":0.7,"pith_summary":"This paper establishes a weighted zero-density estimate for the $L$-functions attached to $\\mathrm{GL}(3)$ Hecke–Maass cusp forms on $\\mathrm{SL}(3,\\mathbb{Z})$. If the theorem is correct, then on average over a spectrally localized family of such forms, essentially no $L$-function has a zero with real part to the right of $1/2+\\sigma$ and imaginary part below $T^{\\theta_1}$, for any fixed $\\sigma>0$ that is not too small. The proof goes through a new asymptotic formula for the twisted second moment of these $L$-functions, proved by means of the $\\mathrm{GL}(3)$ Kuznetsov trace formula, together with a mollified second-moment bound and the argument-principle lemma. The result extends to the spectral aspect of $\\mathrm{GL}(3)$ a line of zero-density estimates previously known for $\\zeta(s)$ and for $\\mathrm{GL}(2)$ forms, and the authors note it can replace the generalized Riemann hypothesis in applications such as the study of $S(t)$ for $\\mathrm{GL}(3)$.","feed_headline":"GL(3) L-functions: zeros off the line are rare on average","feed_subtitle":"A harmonic average over GL(3) cusp forms pushes the zero density down by a power of T.","key_machinery":"The engine of the proof is the Kuznetsov trace formula on $\\mathrm{SL}(3,\\mathbb{Z})$ (Lemma 3.2), applied with the test function $h_{T,M}(\\mu)$ that localizes the spectral parameters of the Hecke–Maass forms to a ball of radius $M=T^{\\theta}$ about the generic point $\\pm\\mu_0$ with $|\\mu_{0,k}|\\asymp T$. This converts spectral averages of $A_j(\\ell_1,\\ell_2)|L(1/2+\\sigma+i\\tau,\\varphi_j)|^2$ into diagonal terms plus Kloosterman and Eisenstein contributions, producing the twisted second-moment asymptotic (Theorem 1.2) with main terms $\\zeta(1+2\\sigma)$ and $\\zeta(1-2\\sigma)$ times the spectral integral, and error terms of sizes $T^{9/4-6\\sigma}M^2$, $T^{2-3\\sigma+\\varepsilon}M^2$, and $T^{83/42-41\\sigma/21+\\varepsilon}M^2(\\ell_1\\ell_2)^{\\vartheta}$. The mollified second moment (Proposition 1.4) is then bounded by $1+O(T^{-2\\sigma\\delta})$ using a mollifier of length $L=T^\\delta$. Finally, the argument-principle lemma (Lemma 4.1) with the box $1/2+1/\\log T \\pm 2iH$ to $3/2\\pm 2iH$ converts this bound into the zero-density estimate, because only the real part of $\\log |LM(s,\\varphi_j)|$ inside the critical strip needs to be controlled.","core_discovery":"Theorem 1.1 is the paper's central claim. For $2/\\log T < \\sigma < 1/2$ and for sufficiently small $\\delta,\\theta_1>0$, the weighted zero count defined in (1.3) obeys $$N(\\$\\sigma$,H) \\ll H $T^{{-\\delta\\sigma}}$\\log T \\qquad\\text{uniformly for }3/\\log T<H<$T^{{\\theta_1}}$.$$ Here $N(\\sigma,H)$ averages, over the $h_{T,M}$-localized harmonic family, the number of zeros with real part exceeding $1/2+\\sigma$ and imaginary part in $(-H,H)$, divided by the box height $H$. The paper's own gloss (Remark 1) is that there are very few $L$-functions in this family with zeros to the right of $1/2+C/\\log T$ and below height $T^{\\theta_1}$.","pith_inferences":["Beyond the paper: the same template—Kuznetsov formula, twisted second moment, mollification, argument principle—should carry over to other automorphic families, such as GL(3) forms of varying level or weight, once the analogous moment asymptotics are available.","Beyond the paper: because the bound is harmonic-weighted, it does not by itself control the proportion of individual forms with an exceptional zero; a non-weighted statement would require an additional large-sieve or spectral-density argument.","Beyond the paper: the uniformity down to $H \\asymp 3/\\log T$ makes the estimate sensitive to the true density of low-lying zeros, so comparing the implied count with the expected number from a GL(3) Weyl law would test whether the exponent $\\delta\\sigma$ has the right shape."],"forward_implications":["For any fixed $\\sigma>0$, the weighted average number of zeros with real part $\\ge 1/2+\\sigma$ and imaginary part below $T^{\\theta_1}$ tends to zero as $T\\to\\infty$, so the family is asymptotically zero-free in that box in the harmonic-average sense.","The bound is uniform in $H$ down to $3/\\log T$, the scale on which the total number of zeros is only $O(\\log T)$, so the estimate remains useful near the critical line.","Letting $\\sigma\\to0$ in the twisted second moment (Corollary 1.3) yields an asymptotic for the harmonic average of $A_j(\\ell_1,\\ell_2)|L(1/2+i\\tau,\\varphi_j)|^2$, a tool for non-vanishing and moment problems for GL(3) L-functions.","As the authors note, the weighted zero-density estimate can replace the generalized Riemann hypothesis in arithmetic applications such as the GL(3) analogue of the $S(t)$ problem."],"supporting_citations":[{"why":"Supplies the test function $h_{T,M}$ and the bounds on the weight functions $\\Phi_{w_4}$, $\\Phi_{w_6}$ used to truncate the Kuznetsov formula.","marker":"[3]"},{"why":"States the spectral Kuznetsov trace formula on $\\mathrm{SL}(3,\\mathbb{Z})$ that turns spectral second moments into Kloosterman sums.","marker":"[7]"},{"why":"Provides the Plancherel-type estimate $H \\asymp T^3 M^2$ and part of the argument for Theorem 7.1.","marker":"[8]"},{"why":"Gives the argument-principle lemma whose real-log feature permits $H$ as small as $1/\\log T$.","marker":"[9]"},{"why":"The GL(2) spectral-aspect zero-density estimate that this paper adapts and extends to GL(3).","marker":"[36]"},{"why":"Supplies the argument-principle lemma (Lemma 4.1) converting mollified second-moment bounds into zero counts.","marker":"[44]"},{"why":"Gives the Kim–Sarnak bound $\\vartheta=7/64$ used in the error terms of the second moment.","marker":"[28]"},{"why":"Provides the GL(3) Kuznetsov applications, including the lemma used in Theorem 7.1 to bound Kloosterman sums.","marker":"[2]"}],"fun_headline_variants":["GL(3) L-function zeros: power drop beyond critical line","Spectral aspect zero-density for GL(3) Hecke-Maass forms","Power saving zero-density from twisted second moment","Sparse GL(3) zeros right of critical line"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the factor $1/H$ in the definition of the weighted zero count and the spectral normalization factor in the mollified second-moment bound are compatible, without showing how the large spectral factor $H_{\\mathrm{spec}}\\asymp T^3M^2$ is removed in the final step of Section 7.","fun_headline_variants_meta":{"raw":{"variants":["GL(3) L-function zeros: power drop beyond critical line","Spectral aspect zero-density for GL(3) Hecke-Maass forms","Power saving zero-density from twisted second moment","Sparse GL(3) zeros right of critical line"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000488,"raw_usage":{"total_tokens":2324,"prompt_tokens":788,"completion_tokens":1536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":404,"completion_tokens_details":{"reasoning_tokens":1465}},"tokens_in":404,"tokens_out":1536,"duration_ms":11256,"temperature":1.0,"reasoning_tokens":1465,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:33:21.103159+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Keep every normalization explicit when Proposition 1.4 is substituted into the argument-principle inequality in Section 7; if the unsuppressed spectral factor $H_{\\mathrm{spec}}\\asymp T^3M^2$ remains after multiplying by the box height $H$, the stated bound $N(\\sigma,H)\\ll H T^{-\\delta\\sigma}\\log T$ cannot follow, and a corrected normalization or a different right-hand side would be needed.","supporting_citations":[{"cited_title":"Blomer and J","cited_arxiv_id":null,"evidence_quote":"Supplies the test function $h_{T,M}$ and the bounds on the weight functions $\\Phi_{w_4}$, $\\Phi_{w_6}$ used to truncate the Kuznetsov formula."},{"cited_title":"Buttcane, The spectral Kuznetsov formula on SLp3q","cited_arxiv_id":null,"evidence_quote":"States the spectral Kuznetsov trace formula on $\\mathrm{SL}(3,\\mathbb{Z})$ that turns spectral second moments into Kloosterman sums."},{"cited_title":"Buttcane and F","cited_arxiv_id":null,"evidence_quote":"Provides the Plancherel-type estimate $H \\asymp T^3 M^2$ and part of the argument for Theorem 7.1."},{"cited_title":"Conrey and K","cited_arxiv_id":null,"evidence_quote":"Gives the argument-principle lemma whose real-log feature permits $H$ as small as $1/\\log T$."},{"cited_title":"Liu and J","cited_arxiv_id":null,"evidence_quote":"The GL(2) spectral-aspect zero-density estimate that this paper adapts and extends to GL(3)."},{"cited_title":"Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2","cited_arxiv_id":null,"evidence_quote":"Gives the Kim–Sarnak bound $\\vartheta=7/64$ used in the error terms of the second moment."},{"cited_title":"Blomer, Applications of the Kuznetsov formula on GLp3q","cited_arxiv_id":null,"evidence_quote":"Provides the GL(3) Kuznetsov applications, including the lemma used in Theorem 7.1 to bound Kloosterman sums."}],"review_version":1}