{"id":"c5d28e36-e075-4350-b7c4-7a3f531c4698","arxiv_id":"2505.18712","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Katz-Sarnak one-level density prediction for Maass form L-functions of prime level now holds for Fourier support up to 15/8, and up to 2 under the Grand Density Conjecture.","lead":"This paper proves that low-lying zeros of L-functions attached to Maass forms of prime level follow the predicted random-matrix statistics for a wider range of test functions than previously known. It extends the Fourier support from 3/2 to 15/8 unconditionally, and to 2 under a standard zero-density conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unconditional 15/8 support rests on Lemma 3.4(ii), a fourth-moment bound for Dirichlet polynomials whose proof is sketched and whose stated t- and d-dependence is the exact point where a wrong constant would break Proposition 3.2 and hence Theorem 1.1.","rationale":"The central claim of the paper is Theorem 1.1, and its proof is a coherent chain of standard tools: explicit formula, Kuznetsov trace formula, reduction to Kloosterman sums, Heath-Brown's identity, and Dirichlet polynomial estimates. I checked the dyadic-splitting lemma (Lemma 3.5), the comparison of error terms against the expanded right-hand side of Proposition 3.2, the support parameter 15/8, and the c/d bookkeeping in Proposition 3.1; these steps appear consistent and the final estimates have enough room. The conditional Theorem 1.3 and the k>2 case of Theorem 1.2 are sketched, but they are not needed for the main unconditional Maass result. The genuinely load-bearing spot is Lemma 3.4(ii), exactly as the reader's weakest_assumption identifies: its d(log dX)^13 bound controls the two exceptional fourth moments in case (b), and if the true d-exponent were d^2 instead of d, the case (b) estimate would exceed the right-hand side of Proposition 3.2 and the 15/8 support would fail. The result is probably true — the fixed-modulus fourth-moment bound is standard and the sketch points to the correct argument — but because the paper does not spell out the application of [Mo1, Theorem 10.1] and the Perron/contour details are asserted, the reader's CONDITIONAL verdict is appropriate. I do not see a reason to move to accept or reject without the missing derivation, and the secondary sketched claims are not central enough to change the verdict. Thus the existing verdict should stand.","tokens_in":23095,"tokens_out":25773,"duration_ms":207307,"concrete_test":"Write out the full proof of Lemma 3.4(ii): state [Mo1, Theorem 10.1] verbatim and verify that it supplies the fixed-modulus bound ∑*_{χ mod d}∫_{|u|≤T}|L(1/2+iu,χ)|^4 du ≪ d T (log dT)^4, not merely the averaged bound over q≤Q. Then recompute the dyadic split in (3.5)-(3.6) with all Perron error terms included. If the fixed-modulus bound is unavailable, replace the fourth-moment input in (3.12) by the available bound and recheck Proposition 3.2 case (b). A useful numerical sanity check is to approximate the left side of Lemma 3.4(ii) for d=3,5 and X=10^2,10^3 and compare its growth to d(log dX)^13, which would detect a grossly wrong d-dependence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is reduced in Proposition 3.1 to Proposition 3.2. In case (b) of Proposition 3.2, the proof applies Lemma 3.4(ii) to the two exceptional indices j1, j2, obtaining a fourth-moment factor with the sharp d-dependence. Lemma 3.4(ii) asserts that, for a(n)=1 or log n, ∫∑*_{χ mod d}|∑_{n≤X}a(n)χ(n)n^{-1/2-it}|^4 dt/(t^2+1) ≪ d(log(dX))^13. The proof of Lemma 3.4(ii) is the least fully written step in the paper: after Perron's formula and a contour shift, it invokes a fourth-moment estimate for Dirichlet L-functions, 'see [Mo1, Theorem 10.1]', and from it concludes the fixed-modulus bound ∑*_{χ mod d}∫_{T0≤|u-t|≤2T0}|L(1/2+α+iu,χ)|^4 du ≪ d(T0+|t|)(log(dX))^4. This fixed-d estimate is exactly what is needed in (3.5)-(3.6); if the cited theorem only supplies the averaged bound over q≤Q, then replacing d by d^2 would make the case (b) estimate (d+P')^{1/2}d, which is not absorbed by the right-hand side of Proposition 3.2. The paper neither states the precise form of [Mo1, Theorem 10.1] nor writes out the Perron errors and horizontal-line contributions. This is the unique place where a plausible but unverified standard input is load-bearing for the main unconditional theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an extended one-level density theorem for low-lying zeros of Maass form L-functions of prime level. Theorem 1.1 establishes the Katz-Sarnak prediction with orthogonal symmetry for even Schwartz test functions whose Fourier transform is supported in (-15/8, 15/8), improving the previous support (-3/2, 3/2) due to Alpoge et al.; Theorem 1.2 gives a parallel improvement for holomorphic newforms, with support Theta_k = 2 - 1/(5k-2); and Theorem 1.3 extends the Maass support to (-2,2) conditionally on the Grand Density Conjecture. The proof follows the ILS strategy: an explicit formula (Lemma 2.1) reduces the density to Hecke eigenvalue sums; a newform-to-full-space comparison (Lemma 2.2) and the Kuznetsov formula (Lemma 2.3) convert these into Kloosterman sums; character orthogonality and Mellin inversion (Proposition 2.6, Lemma 2.7) reduce the problem to weighted character sums of Dirichlet polynomials; Heath-Brown's identity splits the von Mangoldt function into 40 factors; and the resulting mean values are bounded via the large sieve (Lemma 3.3), fourth-moment estimates (Lemma 3.4), and a combinatorial dichotomy (Lemma 3.5) yielding Proposition 3.2. Section 4 adapts the argument to the holomorphic family, and Theorem 1.3 is reduced to the prior work [DFS2].","tokens_in":23444,"tokens_out":64222,"duration_ms":450250,"significance":"Assuming the two technical points flagged below are resolved, the paper is a genuine advance: the unconditional support in the Maass level-aspect family is enlarged by 25% relative to the work of Alpoge et al., and the conditional support (-2,2) is the natural limit of the trace-formula method. The holomorphic improvement of Theorem 1.2, though small, strictly widens the range of [DFS2]. The derivation is parameter-free: no fitted constants or normalization choices enter, and the Katz-Sarnak prediction is tested in an extended range. The paper is largely self-contained, with the large-sieve lemma, the combinatorial splitting lemma, and the Mellin bounds proved in the text. The main burden rests on two passages that are not fully written: the fixed-modulus fourth-moment bound in Lemma 3.4(ii), and the deduction of (3.2) from Proposition 3.2 in Section 3; both are load-bearing for Theorem 1.1, and both appear reparable within the manuscript's scope.","major_comments":[{"comment":"The proof of Lemma 3.4(ii) is the least complete step in the paper. After Perron's formula and the contour shift to Re(s) = alpha, the argument invokes 'an estimate for the fourth moment of Dirichlet L-functions (see for example [Mo1, Theorem 10.1])' and immediately reads off the fixed-modulus bound sum over primitive chi mod d of the integral over T0 <= |u-t| <= 2T0 of |L(1/2+alpha+iu,chi)|^4 du, bounded by d(T0+|t|)(log dX)^4. The paper neither states the precise form of [Mo1, Theorem 10.1] nor writes out the Perron error terms or the horizontal-line contribution. If the cited theorem is the standard average over moduli q <= Q, the fixed-d bound with the stated d-dependence does not follow directly. This d-dependence is load-bearing: in the proof of Proposition 3.2, case (b), the two applications of Lemma 3.4(ii) yield the fourth-root factors that produce the term (d+N^{1/5})^{1/2}; if the correct fourth-moment bound were d^2(T0+|t|)(log dX)^4, the resulting term (d+P')^{1/2}d would not be absorbed by the right-hand side of (3.4), and the saving N^{-epsilon/10} in Proposition 3.1 would fail. The authors should state the exact theorem cited and supply the reduction, including the t-aspect and modulus-aspect bookkeeping.","section":"Section 3, Lemma 3.4(ii), displays (3.5)-(3.6)"},{"comment":"The step just before Proposition 3.2 does not close as written. From Lemma 2.7 one has |Psi(it)| <= N^{15/16-epsilon/2}/((t^2+1)c); combining the stated uniform bound (3.4) with the triangle inequality gives LHS(3.2) << N^{15/16-epsilon/2}(N+d)^2 N^{1/16} d/c, whereas (3.2) claims (c/d)N^{1-epsilon/4}. For c = d = N these are respectively N^{3-epsilon/2} and N^{1-epsilon/4}, so the stated form of Proposition 3.2 is too weak by a factor N^2 (log)^{O(1)} to imply (3.2). The displayed chain Psi(it) << N^{15/16-epsilon/2}/((t^2+1)c) << cN^{15/16-epsilon/2}/((t^2+1)(N+d)^2) is also invalid for c < N+d. The intended implication does go through if one uses the finer per-configuration bounds obtained inside the proof of Proposition 3.2, namely I << d + d^{1/2}N^{9/16-epsilon/200} + N^{15/16-epsilon/2} in case (a) and the analogous case (b) bound, since then N^{15/16-epsilon/2} I/c is at most (c/d)N^{1-epsilon/4}(log dN)^{O(1)} for all c,d in range. The authors should either state the stronger per-configuration bounds as part of Proposition 3.2 or write out the deduction explicitly; the same issue affects the holomorphic reduction in Section 4.","section":"Section 3, passage from Proposition 3.2 to (3.2)"}],"minor_comments":[{"comment":"The abstract spells the trace formula as 'Kutznetsov', and Lemma 4.2 refers to an 'even Schwarz function'; both should read 'Kuznetsov' and 'Schwartz'.","section":"Abstract and Lemma 4.2"},{"comment":"The paper states that Proposition 3.2 is the case k=2 of Proposition 4.3, but the right-hand side of (4.2) at k=2 is (N+d)^2 N^{1/16}/d, whereas (3.4) reads (N+d)^2 N^{1/16} d; these differ by a factor d^2, so the two statements are not consistent and the discrepancy should be resolved.","section":"Section 4, before Proposition 4.3"},{"comment":"The proof of Theorem 1.3 is only a sketch: it refers to [DFS2, Proposition 3.4] and then says 'the rest of the argument is the same as in [DFS2]'. Since Theorem 1.3 is stated as a theorem, the authors should either include the details of the contour shift (which uses Lemma 2.7 in place of [DFS2, Lemma 3.3]) or clearly delimit which parts are imported verbatim.","section":"Section 2, proof of Theorem 1.3"},{"comment":"The definition Theta_k = 2 - 1/(5k-2) is typeset as '2 - 1/5k - 2' without parentheses, which is ambiguous; please fix the typesetting so that the denominator is unambiguous.","section":"Section 1.2, equation (1.6)"}],"recommendation":"major_revision","confidential_remarks":"The two gaps flagged in the major comments are local and appear fixable, so I do not recommend rejection. The referee's main worry is the citation [Mo1, Theorem 10.1]: if it is the averaged fourth moment over moduli q <= Q, the fixed-modulus bound used in Lemma 3.4(ii) requires a supplementary argument, and the editor may wish to ask the authors to address this explicitly. The reduction from (3.2) to Proposition 3.2 also needs rewriting; as stated, the uniform bound (3.4) is not sufficient, although the proof of Proposition 3.2 contains the stronger per-configuration estimates needed. The statement-consistency issue between Proposition 3.2 and Proposition 4.3 at k=2 should also be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real news here is the unconditional admissible support 15/8 for the Maass family, beating Alpoge et al.'s 3/2, plus an improvement in the holomorphic case and a conditional (-2,2) range under the Grand Density Conjecture. The method is genuinely new in this setting: Heath-Brown's identity plus large sieve and fourth moment bounds instead of the usual zero-density route. That is a within-field step forward, not a revolution, but it is a real one.\n\nThe paper does a lot right. The chain from Kuznetsov to the Dirichlet polynomial estimates is coherent, Lemma 3.5's splitting is clean, and the discussion of why the method stops at 15/8 (Remark 3.6) is honest and useful. There is no circularity and no fitted parameters. The self-citations to DFS2 are legitimate prior work, used for the holomorphic analogue.\n\nThe soft spots are real but proportionate. Lemma 3.4(ii), the fourth moment bound with d(log dX)^13, is load-bearing for the main theorem, and its proof is a sketch: it invokes Montgomery's fourth moment estimate without stating it precisely, and the Perron error terms and horizontal line contributions are asserted rather than shown. The stress-test note worries that if this lemma only holds in an averaged form over moduli, the d-dependence would be off by a factor of d, and that would break Proposition 3.2. I cannot rule that out from the text as written, but I also do not find it likely; the claimed fixed-d estimate is exactly the sort of thing that follows from Montgomery's work, and the sketch points the right way. Still, it is the one place where a standard input is doing heavy lifting without full verification, and the authors should be required to write out the details or at least give a precise statement of the cited theorem.\n\nThe other two soft spots are minor. Theorem 1.3 is explicitly a reduction to DFS2, and the k > 2 part of Theorem 1.2 is also a reduction. These are acceptable in a paper whose main result is the Maass 15/8 theorem, but they should be clearly flagged as sketches rather than full proofs.\n\nFor whom: analytic number theorists working on low-lying zeros or the level aspect of families. It deserves a serious referee; the referee should focus on Lemma 3.4(ii) and ask for a complete derivation. I would not block acceptance, but I would want the lemma proven carefully before publication.\n\nRecommendation: yes, send it to peer review, with the expectation of a revision that fills in the fourth moment details.","headline":"Solid extension of the low-lying zeros support in the Maass family from 3/2 to 15/8; the main theorem looks correct, but a load-bearing fourth-moment lemma is only sketched and should be written out.","tokens_in":24062,"tokens_out":1346,"would_cite":true,"duration_ms":13537,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11F72","11M41","11N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an unconditional one-level density theorem for Maass form L-functions of prime level, with Fourier support 15/8 instead of the previous 3/2.","keywords":["low-lying zeros","one-level density","Maass newforms","Kuznetsov trace formula","Dirichlet polynomial averages","large sieve","fourth moment of Dirichlet L-functions","holomorphic modular forms"],"falsifier":"Compute the integral on the left-hand side of Proposition 3.2 for $d=N$, $N_1=\\cdots=N_5=N^{3/8}$, $N_6=\\cdots=N_{40}=1/2$, and $a_j=1$ for $j\\le5$, $a_j=0$ otherwise, so that $N_1\\cdots N_{40}=N^{15/8}$; if for a sequence of primes $N$ the integral exceeds $C N^{17/16}(\\log N)^{O(1)}$ by more than $N^{-\\varepsilon/10}$, then the proof of Theorem 1.1 fails, since Remark 3.6 identifies this exact configuration as the bottleneck.","tokens_in":22883,"feed_emoji":"🔢","tokens_out":13352,"duration_ms":128842,"temperature":0.7,"pith_summary":"This paper proves that the low-lying zeros of Maass form L-functions in the family of prime level $N$ obey the expected random-matrix symmetry for a wider class of test functions than previously known. Specifically, for every even Schwartz function whose Fourier transform is supported inside $(-15/8,15/8)$, the one-level density converges, as $N\\to\\infty$ through primes, to the density $W^{(O)}(x)=1+\\tfrac12\\delta_0(x)$ that describes eigenvalue repulsion near $1$ for large random orthogonal matrices. The previous unconditional range was $(-3/2,3/2)$. The proof routes the density through the explicit formula and the Kuznetsov trace formula into averages of Dirichlet polynomials, and those averages are controlled by the large sieve together with a fourth-moment bound for Dirichlet $L$-functions. This matters because wider support means more low-lying zeros are captured, and density theorems of this type translate into quantitative control of central vanishing on average.","feed_headline":"Density theorem for Maass L-function zeros extends to 15/8","feed_subtitle":"A sharper error-term argument widens the verified support from 3/2 to 15/8, tightening the match to random-matrix symmetry.","key_machinery":"The load-bearing mechanism is a chain of transformations from a spectral average to a character-sum average. After the explicit formula writes the one-level density as a sum of Hecke eigenvalues at prime powers, the Kuznetsov trace formula turns this into a weighted sum of Kloosterman sums; the crucial analytic ingredient is a detailed study of the Bessel kernel $H^+(x)$ in Lemma 2.4, giving its Taylor expansion and derivative bounds near $x=0$. Orthogonality of characters converts the Kloosterman sums into sums over primitive Dirichlet characters, and Heath-Brown's identity decomposes the von Mangoldt function into convolutions of $1$, $\\mu$, and $\\log$. The final object is an integral over $t$ of products of Dirichlet polynomials, and the decisive estimates are a large-sieve second-moment bound and a fourth-moment bound (Lemma 3.4) for such polynomials, whose conductor dependence decides the width of the admissible support.","core_discovery":"The central claim is Theorem 1.1: for an even Schwartz function $\\phi$ with $\\operatorname{supp}(\\hat{\\phi})\\subset(-15/8,15/8)$, any fixed weight $h$, and prime $N$, one has $$D^*(\\phi,h;N)=\\int_{\\mathbb{R}} $W^{{(O)}}$(x)\\$\\varphi$(x)\\,dx+o(1),\\qquad $W^{{(O)}}$(x)=1+\\tfrac12\\delta_0(x).$$ This is the random-matrix prediction for a family with orthogonal symmetry: after scaling zeros by $\\log N/2\\pi$, their low-lying distribution matches the behaviour of eigenvalues near $1$ of random orthogonal matrices. The paper obtains this unconditionally, extending the earlier support $(-3/2,3/2)$, and gives a conditional variant (Theorem 1.3) reaching $(-2,2)$ under the Grand Density Conjecture. It also proves the analogous support extension for holomorphic newforms of even weight (Theorem 1.2), where the admissible support becomes $\\Theta_k=2-1/(5k-2)$.","pith_inferences":["Inference: the proof's dependence on the fourth-moment estimate means that a genuinely stronger fourth-moment bound for Dirichlet $L$-functions would, through the same lemmas, immediately push the unconditional support toward $(-2,2)$ without any new structural idea.","Inference: the bottleneck configuration identified in Remark 3.6, five character sums of lengths near $N^{3/8}$, is the natural test case for any attempt to go beyond $15/8$; a numerical or structural bound for the large values of those sums would have immediate consequences for the density theorem.","Inference: the same reduction should extend to other level-aspect families, since only the trace formula and the Mellin-transform bound for the relevant kernel are family-specific; families whose kernels satisfy Lemma 2.7-type bounds should inherit the $15/8$ support."],"forward_implications":["The Maass newform family of prime level exhibits the expected orthogonal density for all test functions with Fourier support up to $15/8$; the previous unconditional ceiling was $3/2$.","The same method lifts the holomorphic-form level-aspect support to $\\Theta_k=2-1/(5k-2)$ for every even weight $k$, improving the earlier values.","Under the Grand Density Conjecture, the Maass family support extends all the way to $(-2,2)$.","The proof supplies a power saving $N^{-\\varepsilon/10}$ in the main error term, which is exactly the quantitative form needed to derive average upper bounds on vanishing at the central point."],"supporting_citations":[{"why":"Supplies the previous unconditional support $(-3/2,3/2)$ in this Maass family, the baseline Theorem 1.1 extends.","marker":"[A+]"},{"why":"Gives the analogous extension for holomorphic forms and the method Theorem 1.3 follows; Theorem 1.2 improves its support.","marker":"[DFS2]"},{"why":"Sources the Kuznetsov trace formula (Lemma 2.3) and the spectral-weight setup.","marker":"[KL]"},{"why":"Provides the fourth-moment bound for Dirichlet L-functions used to prove the decisive Lemma 3.4(ii).","marker":"[Mo1]"},{"why":"Supplies Heath-Brown's identity, the large-sieve framework, and the Grand Density Conjecture statement.","marker":"[IK]"},{"why":"Gives the explicit formula (Lemma 2.1) connecting the zero sum to Hecke eigenvalues.","marker":"[RS2]"},{"why":"Provides the basis-decomposition technique for relating newform-level sums to level-one sums (Lemma 2.2).","marker":"[ILS]"},{"why":"Provides Perron's formula used in the fourth-moment reduction in Lemma 3.4(ii).","marker":"[Ko]"}],"fun_headline_variants":["Maass L-function zero density support widens to 15/8","Low-lying zero density for Maass forms now proven to 15/8","Maass family L-functions: density theorem extends support to 15/8","Zeros of Maass L-functions: new support bound 15/8","Beyond 3/2: Maass L-function zero density support 15/8"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the fourth-moment estimate in Lemma 3.4(ii) for character sums whose coefficients are $1$ or $\\log n$; that estimate is derived through Perron's formula together with a cited fourth-moment theorem for Dirichlet $L$-functions, and if it is even slightly too optimistic the $N^{-\\varepsilon/10}$ saving in Proposition 3.1 disappears and the $15/8$ support collapses.","fun_headline_variants_meta":{"raw":{"variants":["Maass L-function zero density support widens to 15/8","Low-lying zero density for Maass forms now proven to 15/8","Maass family L-functions: density theorem extends support to 15/8","Zeros of Maass L-functions: new support bound 15/8","Beyond 3/2: Maass L-function zero density support 15/8"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000785,"raw_usage":{"total_tokens":3479,"prompt_tokens":972,"completion_tokens":2507,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":2402}},"tokens_in":588,"tokens_out":2507,"duration_ms":19130,"temperature":1.0,"reasoning_tokens":2402,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:27:31.854006+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the integral on the left-hand side of Proposition 3.2 for $d=N$, $N_1=\\cdots=N_5=N^{3/8}$, $N_6=\\cdots=N_{40}=1/2$, and $a_j=1$ for $j\\le5$, $a_j=0$ otherwise, so that $N_1\\cdots N_{40}=N^{15/8}$; if for a sequence of primes $N$ the integral exceeds $C N^{17/16}(\\log N)^{O(1)}$ by more than $N^{-\\varepsilon/10}$, then the proof of Theorem 1.1 fails, since Remark 3.6 identifies this exact configuration as the bottleneck.","supporting_citations":[],"review_version":1}