{"id":"b4349818-183b-4209-91cc-b03be3faac54","arxiv_id":"2507.14566","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At least 33% of the special values L(1/2+it_f, f) of Hecke-Maass L-functions are nonzero, a new effective non-vanishing record.","lead":"This paper proves that at least one third of special Hecke-Maass L-values, evaluated at the eigenvalue-dependent point 1/2+it_f, do not vanish, improving the known one quarter for central values. Under the Riemann hypothesis the proportion rises to one half, and new short-interval versions are obtained as well.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unconditional 33% theorem depends on the deferred Bessel lemmas 4.1–4.3, especially the asserted H^- extension behind Lemmas 8.2 and 9.3; the rest of the moment and mollifier chain appears internally consistent.","rationale":"Reading in good faith, the paper has a clear architecture: approximate functional equations, twisted first and second moments via Kuznetsov, then a mollifier and Cauchy, then harmonic-weight removal. I checked the mollifier optimization in §10.3–10.4: the constraints Δ<1/2, Δ<(2ν+1)/4, and Δ<(2ν+1)/3 combine to give the optimal proportion min{1/3,(2ν+1)/(2ν+5)}, and the final constants in Theorems 1 and 2 follow. The 'max' in (10.23) is a typo; the displayed (10.28) and (1.5) use the correct 'min', so it is not load-bearing. I also checked the error-term bookkeeping in Lemma 10.2: the potentially problematic T∑(m1,m2)/(m1m2) term is controlled by the square-free support of the mollifier, and the stated constraints are consistent with the claimed error ΠT^{1-ε}. The density-theorem part is conditional on RH and has its own deferred Bessel analysis, but it is not needed for the unconditional 33% claim. The genuine soft spot is therefore exactly the one the reader identified: Lemmas 4.1–4.3 are deferred to the same author's companion paper, and the H^- extension is asserted rather than proved. Those lemmas control the off-diagonal terms in the two twisted moments, and without them the 1/3 lower bound has no independent verification in this text. The concrete test above would settle the H^- question. Since no stronger internal inconsistency emerged, the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":30175,"tokens_out":24761,"duration_ms":292038,"concrete_test":"Verify Lemma 4.3 for H^- rather than H^+. (i) Re-derive (2.8) from Watson 3.7(6) tracking signs and constants, and prove the stationary-phase assertion in Lemma 4.2 for f_-(r;v,w)=v e^r - w e^{-r} using the stationary-phase lemma B.1 of [Qi]; (ii) numerically evaluate the H^- Bessel transform (2.7) for a representative kernel, say h(t)=exp(-((t-T)/Π)^2) with T=10^4, Π=T^{0.6}, y=√p, x=4π√p/c, choosing p≈cT so that v,w≫T, and check the result is O(T^{-10}). If (i) fails or (ii) is not negligible, the c-truncation in §§8.3/9.3 and the error term in Theorem 6 are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central unconditional claim Theorem 1 is reached through the twisted moment asymptotics Theorem 6 and the mollifier calculation in §10. The hardest analytic load in that chain is carried by the off-diagonal bounds Lemmas 8.2 and 9.3. These are derived in §§8.3 and 9.3 by applying Corollary 4.1 to truncate the Kloosterman c-sum at c≪mN/T. Corollary 4.1 is a direct consequence of Lemmas 4.1–4.3, whose proofs are not contained in this paper: the text says the H^+ results are in the same author's companion paper [Qi], and that the H^- kernel (2.7) 'may be extended ... by literally the same proofs'. The H^- case is not a trivial relabeling: it passes through the I-Bessel identity (2.8) and yields the phase f_-(r;v,w)=v e^r - w e^{-r} in Lemma 4.2, so the stationary-phase analysis is not literally identical to the published H^+ analysis. A sign or decay error here would invalidate the c-truncation, the error term in Theorem 6, and hence the 1/3 constant. The §10.4 'max' typo is real but harmless, since the final Theorem 2 and (10.28) correctly use 'min'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves nonvanishing results for Hecke-Maass L-functions at the special point s_f = 1/2 + it_f. The main unconditional theorem (Theorem 1) states that for each parity class B_delta, at least one third of the values L(s_f, f) with t_f <= T are nonzero as T grows, and the same holds on the full basis B. Theorem 2 gives a short-interval version with proportion min{1/3, (2mu+1)/(2mu+5)}. Under the Riemann hypothesis for the relevant L-functions and Dirichlet L-functions, Theorems 3 and 4 raise the long-interval proportion to 1/2 and give mu/(mu+1) for mu > 1/2. The proof combines the Kuznetsov formula, approximate functional equations, Bessel-integral analysis, a mollifier method, and an extended density theorem for the 1-level density of L(s_f, f). The paper is technically dense and the main line of argument is plausible, but several load-bearing estimates are deferred or misstated and need correction.","tokens_in":30439,"tokens_out":16083,"duration_ms":127246,"significance":"If the central claims are correct, this is a substantial advance: it gives the first explicit nonvanishing proportion above the 25% known for central values, does so separately in the even and odd families, and includes effective short-interval refinements. The method is a careful combination of standard tools, and the paper is unusually explicit about the constants and error terms. The density-theorem part identifies the unitary symmetry of the family and is of independent interest. The main caveat is that the unconditional 33% result depends on Bessel-integral estimates whose H^- variant is deferred to a companion paper; this is a genuine load-bearing point. The paper does not provide machine-checked proofs or code, but the analytic structure is transparent enough that verification is feasible.","major_comments":[{"comment":"The H^- extension of the Bessel-integral analysis is asserted to follow \"by literally the same proofs\" from the companion paper [Qi], but the phase in Lemma 4.2 is f_-(r;v,w)=v e^r - w e^{-r}, obtained through the I-Bessel identity (2.8), and the stationary-phase analysis is not literally identical to the H^+ case. Since Corollary 4.1 supplies the c-truncation used in Lemmas 8.2 and 9.3, and hence the error term in Theorem 6, the 1/3 nonvanishing constant depends directly on this deferred analysis. The paper should either include the H^- proofs or give a precise statement of the modifications needed and confirm that the companion paper contains them.","section":"Section 4, Lemmas 4.1-4.3 and Corollary 4.1"},{"comment":"The error term in Lemma 9.2 and Theorem 6 is stated as O(T^epsilon( Pi sqrt(T) + T sqrt(r) + Pi^3/(T sqrt(r)) )), but the derivation leading to (9.12) yields an error of order T^{1+epsilon}/sqrt(r) from the diagonal/off-diagonal split, together with Pi T^{1/2+epsilon} and Pi^3 T^epsilon/(T sqrt(r)). With the printed T sqrt(r), the contribution to the mollified second moment in Section 10.3 would be of size T times a sum of 1/gcd(m1,m2), which is about T M^2; this is incompatible with the constraints in Lemma 10.2 and would block the nonvanishing proportion. The text in Section 10.3 appears to use the correct T/sqrt(r) form, so the statements in Lemma 9.2 and Theorem 6 should be corrected and all later uses checked.","section":"Section 9.2, Lemma 9.2, and Theorem 6"}],"minor_comments":[{"comment":"The displayed optimization uses \"max\" where the argument and the final statements (10.28) and Theorem 2 use \"min\". The printed max would claim a proportion exceeding 1/3 in the range mu > 1/2, which is not what the method proves.","section":"Section 10.4, equations (10.23) and (10.24)"},{"comment":"The phrase \"Lemma (15.1)\" should read \"Lemma 15.1\".","section":"Section 15.2, last sentence"},{"comment":"The passage from the weighted lower bound to the unweighted statements in Theorems 1 and 2 is delegated to the Kowalski-Michel method and a citation to [BHS]. Since Theorem 1 is an unweighted density statement, it would be helpful to spell out the specific lemma or adaptation being used.","section":"Section 10.4, removal of the harmonic weight"},{"comment":"The even case requires a split treatment with two different choices of Re(v) and regularity parameters. This is plausible, but a short explanation of why the two regimes overlap and cover all relevant u would improve readability.","section":"Section 4, Remark 4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends on the companion paper [Qi] for the H^+ Bessel analysis and claims the H^- case by the same proof. Given that a load-bearing lemma is deferred and the paper is part of a two-paper project, the editor may wish to ensure that the companion is publicly available and that the H^- extension is actually verified there. The T sqrt(r) versus T/sqrt(r) discrepancy in Lemma 9.2/Theorem 6 must also be resolved before acceptance; the current statement is inconsistent with the later mollifier argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a real result. The author proves the first effective non-vanishing proportion (33%) for the family L(1/2+it_f, f), raises it to 50% under GRH, and adds short-interval versions. Xu only had an ineffective positive proportion for this family, and the twisted moment asymptotics plus the extended density theorem are new. The work is technically substantial and appears to be carried out carefully.\n\nWhat I like: the moment calculations are organized, the error terms are precisely stated, and the mollifier optimization is standard but executed correctly. The paper is honest about where hypotheses enter, and the conditional results are cleanly separated from the unconditional ones.\n\nThe soft spot is real but manageable: the load-bearing Bessel analysis (Lemmas 4.1–4.3) is deferred to the companion paper [Qi], and the extension to the H^- integral is asserted as “literally the same proofs.” The stress-test note is right that the H^- phase f_-(r;v,w) = v e^r − w e^{-r} differs from the H^+ phase, so the stationary-phase analysis is not literally identical. That does not mean the claim is wrong—the stationary-phase condition is analogous and the resulting truncation is plausible—but it is a verifiability gap. A referee should check it, and the author should either include the proofs or pin the companion paper firmly into the record. The paper does display the actual phases and integral representations in (4.7)–(4.8), so a referee has a concrete starting point.\n\nThe §10.4 “max”/“min” typo is real; the final theorem and (10.28) use the correct “min,” so it is harmless.\n\nI do not see circularity: the self-citations supply auxiliary transforms and Weyl laws, not the non-vanishing result. The citation pattern is appropriate.\n\nMy verdict: this deserves a serious referee. The main theorem is a genuine advance, and the proof structure is sound modulo the deferred Bessel lemmas. I would recommend publication after the deferred analysis is made self-contained or the companion paper is firmly integrated, and after the typo is fixed. It is a good candidate for a reading group, though the density-theorem part is heavier going. Yes, accept for peer review.","headline":"A real result: first effective 33% non-vanishing for special Hecke–Maass L-values, with the main soft spot being the deferred Bessel lemmas that deserve a careful referee check.","tokens_in":30988,"tokens_out":6444,"would_cite":true,"duration_ms":70532,"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":"The paper proves that at least one third of the special Hecke--Maass $L$-values $L(\\tfrac12+it_f,f)$ with $t_f\\le T$ are non-zero, with the proportion rising to one half under the Riemann hypothesis.","keywords":["Maass forms","L-functions","non-vanishing","Kuznetsov trace formula","spectral moments","mollifier","low-lying zeros","density theorem"],"falsifier":"A direct computation for a full spectrum up to large $T$ showing fewer than one third of $L(\\tfrac12+it_f,f)$ non-zero would falsify Theorem 1; since that is currently out of reach, the sharpest available check is numerical verification of the Bessel integrals in Lemmas 4.1--4.3, for example the asymptotic (4.7), because the moment asymptotics and hence the one-third bound rest on them.","tokens_in":29900,"feed_emoji":"","tokens_out":17281,"duration_ms":172625,"temperature":0.7,"pith_summary":"The paper studies the non-vanishing of Hecke--Maass $L$-functions at the special point $s=\\tfrac12+it_f$, the point attached to the Laplace eigenvalue of the cusp form. It proves that among forms with $t_f\\le T$, at least one third of the values $L(\\tfrac12+it_f,f)$ are non-zero, uniformly for even forms, odd forms, and the full orthonormal basis; on short intervals $|t_f-T|\\le T^\\mu$, the unconditional proportion is at least $\\min\\{1/3,(2\\mu+1)/(2\\mu+5)\\}$. These are explicit constants, turning an earlier result that only gave an unspecified positive proportion into an effective one. For comparison, the central values $L(1/2,f)$ are known to be non-vanishing in at least 25\\% of cases and vanish trivially on the odd basis, while the special values treated here escape that obstruction. Under the Riemann hypothesis the long-interval proportion rises to one half, with the curious caveat that RH does not improve the short-interval bound for $\\mu\\le 1/2$.","feed_headline":"At least one third of special L-values do not vanish","feed_subtitle":"Beats the 25% known for central values; on the Riemann hypothesis it reaches 50%.","key_machinery":"The argument is carried by two kinds of spectral averages. First, the twisted first and second moments $C_\\delta^1(m)$ and $C_\\delta^2(m_1,m_2)$ over the even or odd Maass spectrum, weighted by the harmonic weight $\\omega_f$ and a Gaussian in $t_f$. The Kuznetsov trace formula splits these averages into diagonal, Eisenstein, and off-diagonal terms; the off-diagonal terms are sums of Kloosterman sums against Bessel transforms, and the Bessel analysis in Section 4 (with full proofs deferred to the companion paper [Qi]) gives the asymptotics of those transforms, while Poisson summation and standard bounds for Kloosterman sums control the off-diagonal terms. The resulting asymptotic formulae, Theorems 5 and 6, supply explicit main terms: $C_\\delta^1(m)=\\Pi T/(\\pi\\sqrt\\pi)\\delta(m,1)+\\cdots$ and $C_\\delta^2(m_1,m_2)$ with main term $(\\Pi T/(\\pi\\sqrt\\pi r))\\big((\\log T/r+\\gamma_\\delta)\\Sigma(\\mathbf m)-2\\bar\\Sigma(\\mathbf m)\\big)$, where $r=m_1m_2/(m_1,m_2)^2$ and $\\Sigma$ is a divisor sum. Second, a mollifier---a short Dirichlet polynomial whose coefficients are optimized by divisor inversion and the Prime Number Theorem---feeds these moments into Cauchy's inequality, producing the one-third lower bound; an unsmoothing lemma converts the smooth weight to sharp intervals. For the Riemann-hypothesis results, the same spectral machinery is applied to the one-level density at the special point through the explicit formula, and an extended density theorem with support $v(\\mu)=\\min\\{3\\mu,1+\\mu\\}$ is proved using variant Kloosterman sums and an identity that splits an ordinary Kloosterman sum into them.","core_discovery":"On the paper's own terms, the central discovery is that the special values $L(s_f,f)$ with $s_f=\\tfrac12+it_f$ are non-vanishing for an explicit positive proportion of the spectrum, with the same constant on both parities: for $\\delta=0,1$ and for the full basis $\\mathcal B$, the liminf over $T$ of the proportion of $f$ with $t_f\\le T$ and $L(s_f,f)\\ne0$ is at least $1/3$. For short intervals the liminf is at least $\\min\\{1/3,(2\\mu+1)/(2\\mu+5)\\}$, refining an earlier unexplicit result. Assuming the Riemann hypothesis for all $L(s,f)$ and Dirichlet $L$-functions, the long-interval proportion rises to at least $1/2$, and for $1/2<\\mu<1$ the short-interval proportion rises to $\\mu/(\\mu+1)$; for small $\\mu$ the conditional method does not beat the unconditional one, a feature the paper highlights. An addendum improves the conditional non-vanishing of central values as well, giving liminf greater than $9/16$ on the even basis and greater than $15/16$ on the odd basis.","pith_inferences":["The one-third constant is a lower bound supplied by the method, not a prediction of the true proportion; the conditional jump to one half suggests that improving the unconditional off-diagonal and Bessel control is the natural route to larger constants.","The same moment machinery, with the deferred Bessel estimates in place, could plausibly be applied to other families evaluated at their spectral points, such as Rankin--Selberg values connected to cusp-form deformation questions, to replace unspecified proportions with explicit ones.","A numerical check on moderately large $T$ comparing the computed non-vanishing ratio of $L(\\tfrac12+it_f,f)$ with $1/3$, $1/2$, and the short-interval constants would be a testable extension, and a ratio systematically above $1/3$ would support the view that the true proportion is larger.","The saturation of the unconditional bound for $\\mu\\le 1/2$ indicates that on very short intervals the bottleneck is the diagonal--mollifier balance rather than the zero-density range, so subconvexity or refined mollifier choices may be the right extension target."],"forward_implications":["For each parity $\\delta=0,1$ and for the full basis $\\mathcal B$, at least one third of the special values $L(\\tfrac12+it_f,f)$ with $t_f\\le T$ do not vanish as $T\\to\\infty$.","On windows $|t_f-T|\\le T^\\mu$, $0<\\mu<1$, the unconditional non-vanishing proportion is at least $\\min\\{1/3,(2\\mu+1)/(2\\mu+5)\\}$, making the earlier unspecified proportion effective.","Under the Riemann hypothesis for $L(s,f)$ and Dirichlet $L$-functions, the long-interval proportion is at least $1/2$, and for $1/2<\\mu<1$ the short-interval proportion is at least $\\mu/(\\mu+1)$; for $\\mu\\le 1/2$ the conditional bound is weaker than the unconditional $1/3$.","The asymptotic formulae for the twisted first and second moments give the leading-order distribution of $L(\\tfrac12+it_f,f)$ at the special point, which can serve as a quantitative baseline for further statistics of these values.","Because the special values do not vanish trivially on the odd basis, the $1/3$ lower bound applies equally to both parity classes, unlike central values."],"supporting_citations":[{"why":"supplies the Bessel-transform estimates (Lemmas 4.1--4.3) on which the twisted-moment asymptotics and all off-diagonal bounds depend.","marker":"[Qi]"},{"why":"supplies the 25% non-vanishing result for central values that motivates the comparison, and the unsmoothing and weight-removal technology adapted here.","marker":"[BHS, LQ2]"},{"why":"gives the earlier unexplicit non-vanishing proportion at the special point that Theorem 2 turns into an effective constant.","marker":"[Xu]"},{"why":"supplies the mollifier and Cauchy-square method by which the moment asymptotics are converted into non-vanishing lower bounds.","marker":"[IS]"},{"why":"supplies the low-lying-zeros density method, the optimal Fourier pair, and the zero-counting argument used for the Riemann-hypothesis results.","marker":"[ILS]"},{"why":"supplies the method, adapted in [BHS], for removing the harmonic weight without weakening the non-vanishing proportions.","marker":"[KM1]"},{"why":"supplies the Kuznetsov trace formula for even and odd cusp forms that lies at the base of both moment computations.","marker":"[CI]"},{"why":"supplies the variant Kloosterman sums used in the extended density theorem for the Riemann-hypothesis results.","marker":"[IL]"},{"why":"supplies the approximate functional equations and the Poisson summation formula used in the moment evaluations.","marker":"[IK]"}],"fun_headline_variants":["Special L-values: 33% non-vanishing, 50% on RH","Hecke-Maass L-functions: improved non-vanishing to 33%","Non-vanishing for shifted L-values beats central case 33% vs 25%","One third of special Hecke-Maass L-values never vanish","RH lifts non-vanishing proportion from 33% to 50%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the technical estimates for Bessel-function integrals used in the moment computations are correct: Lemmas 4.1--4.3 are deferred to a companion paper, and the odd-integral extension is stated without full detail, so the one-third proportion stands or falls with that analysis.","fun_headline_variants_meta":{"raw":{"variants":["Special L-values: 33% non-vanishing, 50% on RH","Hecke-Maass L-functions: improved non-vanishing to 33%","Non-vanishing for shifted L-values beats central case 33% vs 25%","One third of special Hecke-Maass L-values never vanish","RH lifts non-vanishing proportion from 33% to 50%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000424,"raw_usage":{"total_tokens":2216,"prompt_tokens":1030,"completion_tokens":1186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":1079}},"tokens_in":646,"tokens_out":1186,"duration_ms":11209,"temperature":1.0,"reasoning_tokens":1079,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:53:48.339219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computation for a full spectrum up to large $T$ showing fewer than one third of $L(\\tfrac12+it_f,f)$ non-zero would falsify Theorem 1; since that is currently out of reach, the sharpest available check is numerical verification of the Bessel integrals in Lemmas 4.1--4.3, for example the asymptotic (4.7), because the moment asymptotics and hence the one-third bound rest on them.","supporting_citations":[],"review_version":1}