Low-lying zeros in families of Maass form L-functions: an extended density theorem

Anders S\"odergren; Daniel Fiorilli; Kaisa Matom\"aki; Lucile Devin; Martin \v{C}ech

REVIEW 2 major objections 4 minor 37 references

Low-lying zeros in families of Maass form L-functions: an extended density theorem

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.18712 v1 pith:TX7DSXDP submitted 2025-05-24 math.NT

Martin Čech , Lucile Devin , Daniel Fiorilli , Kaisa Matomäki , Anders Södergren This is my paper · ORCID

classification math.NT MSC 11M2611F7211M4111N35

keywords low-lyingzerosone-leveldensityMaassnewformsKuznetsovtraceformulaDirichletpolynomialaverageslargesievefourthmomentofL-functionsholomorphicmodularforms

topics The Riemann Hypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\Theta_k=2-1/(5k-2)$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

The 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.

For 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.

Recommendation: yes, send it to peer review, with the expectation of a revision that fills in the fourth moment details.

Referee Report

2 major / 4 minor

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].

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 (2)

[Section 3, Lemma 3.4(ii), displays (3.5)-(3.6)] 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 3, passage from Proposition 3.2 to (3.2)] 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.

minor comments (4)

[Abstract and Lemma 4.2] 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 4, before Proposition 4.3] 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 2, proof of Theorem 1.3] 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 1.2, equation (1.6)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main unconditional and conditional results are derived from the Kuznetsov trace formula, Heath-Brown's identity, and standard external estimates, with the cited prior papers and the Grand Density Conjecture used only as independent inputs, not as hidden redefinitions of the conclusion.

full rationale

The derivation chain is explicit and self-contained relative to standard external tools: the explicit formula (Lemma 2.1), the reduction to Kloosterman sums (Lemma 2.5 and Proposition 2.6), Heath-Brown's identity, the dyadic decomposition, and the Dirichlet polynomial estimates (Lemmas 3.3 and 3.4). The target support 15/8 appears in Theorem 1.1 and in the hypotheses of Proposition 3.1 and the product condition (3.3), but it is not introduced as a fitted parameter or as an assumption inside the estimates that produce the bound; the final comparison to the Katz-Sarnak integral is a genuine asymptotic statement. Lemma 3.4(ii), the most compressed step, cites Montgomery's fourth-moment theorem [Mo1, Theorem 10.1] as an external input; even if the surrounding argument were insufficiently detailed, that is a correctness or verification risk, not a circular reduction, because the cited theorem is not the paper's own result and does not encode the one-level density conclusion. The cited [DFS2] papers, with overlapping authors, are used for the holomorphic analogue and for the layout of the conditional argument, but those are prior independent works and their use does not assume the truth of the current theorems. The Grand Density Conjecture is explicitly labeled a conjecture and is invoked only for the conditional Theorem 1.3. There are no fitted inputs, no parameters whose values force the prediction, no reliance on the authors' own uniqueness claims, and no renaming of a known result as a new one. Hence the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new postulated entities. Its contribution is a derivation whose external inputs are classical theorems of analytic number theory, plus the explicitly conditional Grand Density Conjecture.

assumptions (6)

domain assumption Grand Density Conjecture as stated in (1.7)
Used only for Theorem 1.3, the conditional extension of support to (-2,2). It is a standard conjecture from Iwaniec-Kowalski, not proved here, and marked as an assumption.
standard math Kuznetsov trace formula, as in [KL] Theorems 7.14 and 8.1
Invoked in Lemma 2.3 to express the spectral average as diagonal, Eisenstein, and Kloosterman terms for the class H(A,delta).
standard math Kim-Sarnak bound (1.2): |alpha_f(p)|, |beta_f(p)| <= p^{7/64}
Deep proven input used in Lemma 2.1 to discard prime powers with nu >= 3 and in Lemma 2.5 for oldform contributions.
standard math Fourth moment bound for Dirichlet L-functions, cited as [Mo1, Theorem 10.1]
Central input in Lemma 3.4(ii), providing the d(log(dX))^13 bound after t-integration. The paper sketches a dyadic reduction but relies on Montgomery's theorem for the fourth moment itself.
standard math Large sieve inequality in the form of Lemma 3.3 and Heath-Brown's identity from [IK, Proposition 13.3]
Used throughout Section 3 to decompose the von Mangoldt function and to factor Dirichlet polynomials into manageable pieces.
standard math Explicit formula for Maass L-functions from [RS2, Proposition 2.1] and the functional equation (1.3)
Basis of Lemma 2.1, converting zero sums into sums of Hecke eigenvalues against the test function.

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Pith. "Pith review of Low-lying zeros in families of Maass form L-functions: an extended density theorem." pith.science (2026). https://pith.science/paper/TX7DSXDP

@misc{pith2026250518712,
  author       = {Pith},
  title        = {Pith review of: Low-lying zeros in families of Maass form L-functions: an extended density theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TX7DSXDP}},
  note         = {Machine review of arXiv:2505.18712}
}

abstract

We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

L.\ Alpoge, N.\ Amersi, G.\ Iyer, O.\ Lazarev, S.\ J.\ Miller, L.\ Zhang, Maass waveforms and low-lying zeros, Analytic number theory: In Honor of Helmut Maier's 60th Birthday , 19--55, Springer, Cham, 2015

work page 2015

[2] [2]

L.\ Alpoge, S.\ J.\ Miller, Low-lying zeros of Maass form L -functions, Int.\ Math.\ Res.\ Not.\ IMRN 2015, no.\ 10, 2678--2701

work page 2015

[3] [3]

S.\ Baluyot, V.\ Chandee, X.\ Li, Low-lying zeros of a large orthogonal family of automorphic L -functions, preprint, arXiv:2310.07606

arXiv

[4] [4]

L.\ Devin, D.\ Fiorilli, A.\ S\"odergren, Low-lying zeros in families of holomorphic cusp forms: the weight aspect, Q.\ J.\ Math.\ 73 (2022), no.\ 4, 1403--1426

work page 2022

[5] [5]

L.\ Devin, D.\ Fiorilli, A.\ S\"odergren, Extending the unconditional support in an Iwaniec--Luo--Sarnak family, preprint, arXiv:2210.15782

work page Pith review arXiv

[6] [6]

S.\ Drappeau, K.\ Pratt, M.\ Radziwi , One-level density estimates for Dirichlet L -functions with extended support, Algebra Number Theory 17 (2023), no.\ 4, 805--830

work page 2023

[7] [7]

W.\ Duke, J.\ B.\ Friedlander, H.\ Iwaniec, The subconvexity problem for Artin L -functions, Invent.\ Math.\ 149 (2002), no.\ 3, 489--577

work page 2002

[8] [8]

E.\ Fouvry, E.\ Kowalski, P.\ Michel, Algebraic trace functions over the primes, Duke Math.\ J.\ 163 (2014), no.\ 9, 1683--1736

work page 2014