{"id":"f88c65d3-2921-413d-801d-a753bc68a635","arxiv_id":"2506.08546","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At least γ(Q)/11 of the L-values L(1/2+it_j, Q⊗u_j) with t_j ≤ T are nonzero, with an analogous short-interval bound for 3/4<μ<1.","lead":"This paper proves that a positive, explicitly computed proportion of certain Rankin–Selberg L-functions do not vanish at special points, improving earlier non-vanishing theorems. The explicit constant is tied to the modular form Q, and the result also holds on short spectral intervals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorems 1 and 2 are supported by the paper's own twisted-moment proofs; the reader's flagged spectral large sieve is not load-bearing for the central claim.","rationale":"The reader's weakest_assumption identified the spectral large sieve (8.3) as the load-bearing input. On inspection, this inequality is not used in the proofs of the headline positive-proportion results: Theorems 1 and 2 are obtained from the twisted first and second moment asymptotics (Theorems 4 and 5), which are proved self-contained within the paper. The large sieve only enters Theorem 8, which is used for the secondary Theorem 3 and in some unsmoothing contexts, and is not needed for the constant γ(Q)/11. The one step that is genuinely load-bearing and not fully demonstrated in the text is the final removal of the harmonic weight ω_j, which is justified by a citation to Kowalski-Michel and BHS. Since this is a standard argument that preserves explicit constants, it does not constitute a significant objection. I found no internal contradictions, no post-hoc exclusions, and no fitted constants in the main derivation; the arithmetic constant γ(Q) in (1.5) correctly matches the ratio γ5/γ1 of residues defined in (10.24) and (3.33). Thus the reader's ACCEPT verdict stands, with agreement only partial because the specific risk the reader emphasized is not the true weak point of the paper.","tokens_in":44026,"tokens_out":21093,"duration_ms":224915,"concrete_test":"Trace the dependency graph: verify that Lemma 10.1 and Lemma 10.3 cite only Theorems 4 and 5, and that the proof of Theorem 5 nowhere invokes (8.3) or Theorem 8. Then write out the Kowalski-Michel/BHS weight-removal argument explicitly, starting from the weighted lower bound (10.37), and confirm that it yields unweighted density γ(Q)/11 - ε without any loss of the explicit constant. If both checks pass, the central claim is fully supported as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central nonvanishing theorems (Theorems 1 and 2) follow from the mollified-moment Lemmas 10.1 and 10.3, which apply Theorems 4 and 5. Theorem 5 is proved in Sections 6-7 using the Kuznetsov formula, Voronoi summation, the Wilton bound, and Luo's identity; the proof does not import the spectral large sieve (8.3) or Theorem 8. The large sieve appears only in Theorem 8, which supports the weaker short-interval Theorem 3 and some unsmoothing applications, but not the positive-proportion results. The final step converting harmonic-weighted nonvanishing to unweighted nonvanishing is delegated to the standard Kowalski-Michel/BHS weight-removal method; this is a citation rather than a proof, but the argument is standard and preserves explicit constants. No internal inconsistency or concrete error surfaced in the main chain, though Section 7.3 is extremely technical and could not be fully checked line-by-line. Residual risk is low and does not justify changing the reader's acceptance.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an effective positive-proportion non-vanishing result for the Rankin--Selberg L-values L(1/2+it_j, Q⊗u_j), where Q is a holomorphic Hecke newform of square-free level and u_j are Hecke--Maass cusp forms of full level. Theorem 1 states that, as T→∞, the unweighted proportion of non-vanishing values among 0<t_j≤T is at least γ(Q)(1/11−ε), with γ(Q) given by an explicit Euler product in (1.5). Theorem 2 gives the analogous lower bound γ(Q)(4μ−3)/(4μ+7)−ε on short intervals |t_j−T|≤T^μ for 3/4<μ<1, and Theorem 3 gives a weaker power bound for 1/3<μ≤3/4. The proof develops asymptotic formulae for twisted first and second spectral moments (Theorems 4 and 5), obtains a mollified first and second moment (Lemmas 10.1 and 10.3), and converts the harmonic-weighted lower bound into an unweighted statement by the standard Kowalski--Michel/BHS weight-removal argument. An additional section derives unsmoothed moment asymptotics (Theorem 11) and a determination theorem (Theorem 10).","tokens_in":44189,"tokens_out":20166,"duration_ms":248355,"significance":"If the proof is correct, this is a substantial quantitative advance over Luo's earlier qualitative positive-proportion result: the proportion is given by an explicit, Q-dependent Euler product, and the method also yields short-interval non-vanishing and improved unsmoothed moment asymptotics. The main chain of the argument is coherent: the central Theorems 1 and 2 follow from the mollified-moment asymptotics, which in turn rest on the twisted-moment theorems proved in Sections 6--7 rather than on the spectral large sieve used only for Theorem 3. The paper is careful with constants and derives γ(Q) from residues of Rankin--Selberg L-functions, with no fitted free parameters. The twisted-moment analysis is long and technical, but the structure of lemmas and error terms is clear, and the external inputs (Kuznetsov formula, Voronoï summation, Luo's identity, and the standard weight-removal method) are standard in the field.","major_comments":[],"minor_comments":[{"comment":"The passage from the harmonic-weighted lower bound (10.37)--(10.38) to the unweighted statements of Theorems 1 and 2 is delegated to the method of Kowalski--Michel [KM1] and its Maass-form adaptation in [BHS]. Since the precise constant γ(Q)/11 is a headline feature, the authors should either state the exact weight-removal lemma they are invoking or give a precise reference to the theorem in [BHS] that preserves the constant, so that the reader does not have to infer it from the literature.","section":"§10.4, final paragraph"},{"comment":"The proof of the off-diagonal second moment says that only the generic case (c,k1k2)=1 will be treated and that the remaining To-type exponential sums are 'not hard' in view of Lemma 7.7. For a paper of this length and with this level of detail elsewhere, a few sentences or a short appendix describing the non-generic cases would greatly help the reader verify that the bound in Lemma 7.12 indeed covers all c, k1, k2.","section":"§7.3.5"},{"comment":"The spectral large sieve (8.3) is quoted from Luo and Jutila without proof. This is acceptable as an external input, but the paper should state more explicitly in Section 8 that this input is used only for Theorem 3 and for the mean Lindelöf bound (2.8), and not for the central positive-proportion Theorems 1 and 2.","section":"§8.1"},{"comment":"The stated coefficient bound x_m=O(τ(m)) does not strictly follow from (10.26), since 1/ξ(m) can be as large as 3^{ω(m)} for square-free m with many small prime factors. All later estimates in the paper only require x_m=O(T^ε) for m≤M, so the statement of (10.3) should be adjusted accordingly.","section":"§10.3, (10.3)"},{"comment":"The notation '3√' (or '3 a') for the cube root in the error term of (2.3) is easy to misread as a product involving a factor 3; the authors should explicitly define the cube-root notation at its first occurrence.","section":"Theorem 4 and §6.4"}],"recommendation":"accept","confidential_remarks":"The paper is technically dense, and I did not verify every exponential-sum estimate in Section 7.3 line by line. The non-generic cases in §7.3.5 are asserted rather than written out, but I did not find a concrete error there or in the main mollifier chain. If the journal can obtain a second referee with expertise in spectral methods, that would be valuable; otherwise, I would recommend asking the author to expand the weight-removal and non-generic-case remarks in revision, without treating them as blocking issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine improvement over Luo's qualitative positive proportion. Theorem 1 gives an explicit Q-dependent constant gamma(Q)/11, and Theorem 2 adds a short-interval version with the proportion (4mu-3)/(4mu+7). The twisted spectral moment asymptotics in Theorems 4 and 5 are new, and the Euler product constant is parameter-free, not fitted to the result. The stress-test note is right: the central chain (mollified moments via Theorems 4 and 5) does not import the spectral large sieve. That sieve only appears in Theorem 8, which supports the weaker short-interval Theorem 3 and some unsmoothing applications. So the reader's flagged weakness is a side issue relative to the headline result.\n\nWhat the paper does well: it is a serious piece of work in the genre. The proofs for the new estimates are given in full, the explicit constants are tracked carefully, and the comparison with Luo's smoothing method is honest and useful. The off-diagonal analysis in Section 7 is the real engine, and it is intricate. I did not check every exponential sum estimate line by line, and I do not think anyone could from one reading, but the structure is coherent, the error terms fall in the right ranges, and the main terms plausibly follow from the stated lemmas.\n\nSoft spots, in proportion. First, several load-bearing inputs are cited rather than reproved: Luo's large sieve, Voronoi summation, and the Kowalski-Michel/BHS weight removal. That is standard in this literature, but it means the final theorem's correctness depends on external results. Second, the weight removal is delegated to a citation rather than shown; if that citation has hidden hypotheses, the explicit constant could be affected. Third, gamma(Q) is an infinite product that is never evaluated for a concrete Q. For all the paper tells us, the proportion could be very small. That is not a flaw in the theorem, but a numerical sanity check for a weight-4 form would have been welcome.\n\nI do not disagree with the reader's ACCEPT verdict; medium correctness risk is about right. This paper deserves a serious referee. Send it to refereeing.","headline":"Solid effective refinement of Luo's non-vanishing theorem; the main new results are genuine and the cited large sieve is not load-bearing.","tokens_in":44712,"tokens_out":2053,"would_cite":true,"duration_ms":24279,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"A positive proportion of Rankin–Selberg special L-values, governed by an explicit Euler product, are shown not to vanish.","keywords":["Rankin–Selberg L-functions","non-vanishing","spectral moments","Kuznetsov trace formula","mollification","Hecke–Maass cusp forms","Euler product","short intervals"],"falsifier":"Compute numerically, for a fixed Q of small level such as an elliptic curve newform, the proportion #{j: t_j ≤ T, L(1/2+it_j, Q⊗u_j) ≠ 0} / #{j: t_j ≤ T} for increasingly large T; if it ever drops below γ(Q)(1/11−ε), Theorem 1 is false. More directly, test the spectral large sieve inequality (8.3) with a_n=1 for N ~ T on spectral intervals of length 1; a counterexample at that level would destroy the proof of Theorem 8.","tokens_in":43796,"feed_emoji":"🔢","tokens_out":6286,"duration_ms":67772,"temperature":0.7,"pith_summary":"This paper proves that, for any fixed holomorphic Hecke newform Q of square-free level, a positive proportion of the Rankin–Selberg L-values attached to the Hecke–Maass spectrum do not vanish at their special points: at least γ(Q)/11 of the values L(1/2+it_j, Q⊗u_j) with t_j ≤ T are nonzero as T→∞, where γ(Q) is an explicit Euler product depending only on Q. On short spectral intervals |t_j−T| ≤ T^μ with 3/4<μ<1, the non-vanishing proportion is at least γ(Q)(4μ−3)/(4μ+7−ε). These are effective, form-dependent positive-proportion results, refining earlier asymptotic first-moment and qualitative positive-proportion results. The proof works through asymptotic formulae for the twisted first and second spectral moments of these L-values, using the Kuznetsov trace formula, Voronoï summation, stationary-phase analysis, and a Selberg-type mollifier whose coefficients are chosen optimally.","feed_headline":"At least γ(Q)/11 of special L-values never vanish","feed_subtitle":"New effective bounds give a form-dependent density of non-vanishing special values as T grows.","key_machinery":"The central objects are the twisted spectral moments C_1(m) and C_2(m_1,m_2) with Gaussian weight exp(−(t_j−T)^2/$Π^{2}$), evaluated through the Kuznetsov trace formula, the Voronoï summation formula, and stationary-phase analysis of the resulting Bessel integrals. The first moment has its main term only at m=1; the second moment carries the main term through the Rankin–Selberg L-function L(s,Q⊗Q) expanded near its pole at s=1, producing the constants γ_1, γ_0 and the factor B(m). The mollifier M_j = Σ_{m≤M} x_m a(m) λ_j(m) $m^{{−1/2−it_j}}$ is chosen by optimizing a Rayleigh quotient with coefficients x_m, and the key input controlling the second mollified moment is the mean Lindelöf bound obtained from Luo's spectral large sieve inequality.","core_discovery":"This paper establishes that, for a fixed holomorphic Hecke newform Q of square-free level q, the special Rankin–Selberg L-values L(1/2+it_j, Q⊗u_j) — where the u_j run over an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue 1/4+$t_j^{2}$ — vanish less often than previously known: at least γ(Q)(1/11−ε) of the values with t_j ≤ T are nonzero as T→∞, with γ(Q) equal to the explicit Euler product in (1.5) built from the Hecke eigenvalues a(p). For short windows |t_j−T| ≤ T^μ with 3/4<μ<1, the non-vanishing proportion is at least γ(Q)(4μ−3)/(4μ+7−ε). The proof proceeds by asymptotically evaluating the twisted first and second spectral moments of these L-values, then applying a Selberg-type mollifier to convert the moment information into a counting lower bound.","pith_inferences":["A natural testable extension is to evaluate γ(Q) numerically for a few concrete newforms (for instance, the Δ form or a CM form) and compare the observed non-vanishing proportion with γ(Q)/11; the paper itself notes that γ(Q) requires numerical evaluation.","The short-interval constant (4μ−3)/(4μ+7) is small even at μ close to 1; pushing the method below μ=3/4 would likely require a genuinely new bound for the second spectral moment rather than a sharper analysis of the present argument.","Theorem 10 suggests that the twisted second moment at the points t_j is a complete invariant for the newform Q; a stronger version might establish the same conclusion using only a finite set of exceptional j.","The load-bearing role of the spectral large sieve inequality indicates that any improvement of that inequality with N ~ T would translate directly into an improvement of the 1/11 proportion."],"forward_implications":["For each fixed Q, the lower bound γ(Q)/11 is explicit and computable from the Hecke eigenvalues a(p), so the result yields a concrete numerical proportion for any given newform.","On short spectral intervals of length about T^μ with μ>3/4, the non-vanishing proportion stays bounded below by γ(Q)(4μ−3)/(4μ+7), so the phenomenon is not confined to the full spectral range.","The asymptotic for the untwisted second spectral moment gives a power-saving remainder, refining Luo's smoothed moment asymptotics.","In the Phillips–Sarnak deformation picture, a positive proportion of non-vanishing special values is exactly what forces the Weyl law to fail for generic co-finite groups under the standard eigenvalue multiplicity assumptions, as recalled in the introduction.","The same moment asymptotics determine the form Q: if |L(s_j,Q⊗u_j)| = c |L(s_j,Q'⊗u_j)| for all j with fixed c>0 then Q=Q'."],"supporting_citations":[{"why":"Supplies the spectral Kuznetsov trace formula that converts the spectral moments into exponential sums.","marker":"[Kuz]"},{"why":"Supplies the Voronoï summation formula and the Euler product structure for the Rankin–Selberg L-functions.","marker":"[KMV]"},{"why":"The Deligne bound |a(n)| ≤ τ(n) is used to control the Euler factors and error terms.","marker":"[Del]"},{"why":"Proves the spectral large sieve inequality (8.3), which yields the mean Lindelöf bound for the untwisted second moment.","marker":"[Luo3]"},{"why":"Independent proof of the same spectral large sieve inequality (8.3).","marker":"[Jut]"},{"why":"The earlier positive-proportion result via mollification that this paper refines.","marker":"[Luo5]"},{"why":"The Selberg-type mollification technique used to pass from moment asymptotics to non-vanishing counts.","marker":"[IS]"},{"why":"Provides the mollifier coefficient choice and the lower-bound method for the rank, adapted to the Maass-form setting.","marker":"[KM2]"},{"why":"The unsmoothing lemma behind Lemma 9.1, which removes the Gaussian weight and yields the short-interval statements.","marker":"[IJ]"},{"why":"The adapted version of the unsmoothing lemma used in Lemma 9.1.","marker":"[LQ]"}],"fun_headline_variants":["γ(Q)/11 of special L-values don't vanish","New non-vanishing bound for Rankin-Selberg L-values","Explicit non-vanishing proportion for special L-values","Improved non-vanishing density in short intervals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the spectral large sieve inequality (8.3): that sums over nearby Maass forms of arbitrary coefficient vectors are as small as (T+N)(TN)^ε, and if that inequality fails, the mean Lindelöf bound and the entire mollified second-moment control collapse.","fun_headline_variants_meta":{"raw":{"variants":["γ(Q)/11 of special L-values don't vanish","New non-vanishing bound for Rankin-Selberg L-values","Explicit non-vanishing proportion for special L-values","Improved non-vanishing density in short intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000889,"raw_usage":{"total_tokens":3841,"prompt_tokens":957,"completion_tokens":2884,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":2816}},"tokens_in":573,"tokens_out":2884,"duration_ms":27132,"temperature":1.0,"reasoning_tokens":2816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:07:09.236356+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute numerically, for a fixed Q of small level such as an elliptic curve newform, the proportion #{j: t_j ≤ T, L(1/2+it_j, Q⊗u_j) ≠ 0} / #{j: t_j ≤ T} for increasingly large T; if it ever drops below γ(Q)(1/11−ε), Theorem 1 is false. More directly, test the spectral large sieve inequality (8.3) with a_n=1 for N ~ T on spectral intervals of length 1; a counterexample at that level would destroy the proof of Theorem 8.","supporting_citations":[],"review_version":1}