Bounding The Number of Zeros Near the Central Point in Families of Cuspidal Newforms

Christopher Housholder; Devayani Pradhan; Joshua Khan; Lucas Chen; Steven J. Miller

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Bounding The Number of Zeros Near the Central Point in Families of Cuspidal Newforms

Lucas Chen, Christopher Housholder, Joshua Khan, Steven J. Miller, Devayani Pradhan

Number Theory · Submitted 15 Aug 2026 · 25 references

Pith's verdict: New counting bound for low-lying zeros (N^{7/8} log N) is sound; positive proportion theorem rests on an unverified moment inequality the abstract overstates.

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CitationChristopher Housholder, Devayani Pradhan, Joshua Khan, Lucas Chen, Steven J. Miller. "Bounding The Number of Zeros Near the Central Point in Families of Cuspidal Newforms." arXiv:2608.15063 (2026). https://pith.science/paper/EP3C6VGR

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Claim paper

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Editor's note

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The paper gives two genuinely new counting results for low-lying zeros in families of even cuspidal newforms of fixed weight and prime level. Theorem 1.15 is a quantitative lower bound: assuming GRH, if the 1-level density inequality holds, then limsup a_{omega,N}/(N^{7/8} log N) > 0. The proof is clean: use a nonnegative test function phi supported small enough to bound b_{omega,N} by O(N^{1/8}/log N), then combine with the product lower bound from the centered moment argument. That part holds up; the details are fine and the constants are explicit enough.

The second main result, Theorem 1.18, aims for a positive proportion. The Cauchy-Schwarz averaging argument with Lemma 3.2 is a real technique, and the 2n-th moment bound for the zero count is useful for the upper-tail corollaries. But the theorem is conditional on an unverified strict inequality: there must exist an odd n >= 3 and a window omega with (-mu)^n < S(n,a;phi_omega). The paper never shows such a triple exists and gives no numerical evidence. The only numerical anchor, omega_min about 0.21864 from [Aro+25], verifies the n=1 condition (1.38), not the n >= 3 condition. As written, if that inequality fails for all odd n >= 3, Theorem 1.18 and its corollaries are vacuous. The abstract's positive-proportion claim overstates the result; it should say conditional on an unverified moment inequality, or the authors should verify it for some explicit parameters.

There are also minor notational issues: sigma is overloaded in (2.16), and the replacement of finite-level mean by limiting mean is delegated to [Aro+25], which is fine but should be more explicit. The reliance on Theorem 1.8 and [Aro+25] is not circular: both are published peer-reviewed results and the new counting argument stands independently. The citation pattern is appropriate.

Who is this for: researchers in low-lying zeros and families of L-functions. It deserves a serious referee; the referee should push for verification or explicit conditioning of the positive proportion claim, and a revised abstract. I would also suggest the authors check whether the n=1 case can be pushed to give a positive proportion via a different route, or at least report the status of the n >= 3 inequality.

Referee report

Major objections (3)

[Section 3.3, Theorem 1.18] The hypothesis (−μ)^n < S(n,a;ϕ_ω) with n ≥ 3 odd is the load-bearing premise for all positive-proportion claims, but it is never verified for any explicit parameters. Remark 1.16's ω_min ≈ 0.21864 is tied to Theorem 1.15's condition (1.38), i.e. to the n = 1 one-level density S(1,σ;ϕ_ω), not to the centered-moment term S(n,a;ϕ_ω) for n ≥ 3. Please supply a numerical or analytic verification for at least one triple (n,a,ω,h), or explicitly state that the positive-proportion result is conditional on an unverified inequality and adjust the abstract accordingly.
[Section 3.3, Eqs. (3.17)-(3.18)] The lower bound liminf (1/|F_N|)∑(X_f−μ)^n ≥ S(n,a;ϕ_ω) requires passing from the finite-level mean m_N in the centered-moment theorem to the limiting mean μ. The sketch in (3.18) needs to be expanded: one must show that the lower centered moments (D(f;ϕ_ω)−m_N)^j are uniformly bounded for j ≤ n under the support condition, so that the terms involving m_N−μ vanish. As written, this step is delegated rather than proved.
[Theorem 1.18 statement] The integer a in S(n,a;ϕ_ω) is not quantified in the theorem statement. The hypothesis should read "there exist an integer a with 0 ≤ a ≤ n/2 and a window ω such that supp(ϕ̂_ω) ⊂ (−1/(n−a), 1/(n−a)) and (−μ)^n < S(n,a;ϕ_ω)". Without this quantification the hypothesis is ill-posed.

Minor objections (5)

[Section 2, Eq. (2.6)] The displayed expression appears garbled; it should be −μ^n + (a_{ω,N}/|F_N|)μ^n + (1/|F_N|)∑_{f∈eF_N}(∑_{|γ|≤ω}ϕ_ω(γ)−μ)^n.
[Section 2, Eq. (2.16)] The symbol σ is overloaded. The Fourier support of the displayed φ is [−σ/2, σ/2], so the claimed support in [−1/2, 1/2] holds only after setting σ = 1; please rename the parameter or state σ = 1 explicitly.
[Definitions 1.14 and 1.17] The boundary convention is inconsistent: R_f(ω) is defined with the closed interval [−ω,ω], whereas eF_N and X_f effectively use the open window because ϕ_ω vanishes at ±ω. This can make forms with a zero exactly at ±ω counted by R_f but not by the positive-proportion argument; align the conventions.
[Section 1.1] The phrase "on obtains" should be "one obtains".
[References] The references [Mon73a] and [Mon73b] are the same article, as are [RS96a] and [RS96b]; these duplicates should be merged.

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The paper

The paper claims

Under GRH, a positive proportion of even cuspidal newforms of prime level has a zero in a prescribed window about the central point whenever a stated centered-moment inequality holds.

The reading

Earlier work on even holomorphic cuspidal newforms of fixed weight and prime level showed that at least one form in the family has a zero in a normalized window $(-\omega,\omega)$ about the central point, for $\omega$ as small as about $0.21864$. This paper asks how widespread that phenomenon is, and its aim is to prove that such zeros are common rather than exceptional. Assuming the Generalized Riemann Hypothesis, it shows that under a first-moment inequality the number of forms with a zero in the window grows at least like $N^{7/8}\log N$ along a sequence of prime levels, and that under an odd-moment inequality a positive proportion of the whole family has such a zero. It further proves that forms with many zeros in the window are rare, and that some fixed positive number of zeros occurs on a positive proportion of forms. These are quantitative statements about how the arithmetic information carried by low-lying zeros is spread across the family, going beyond the earlier guarantee that a zero exists somewhere.

What carries the argument

The machinery is the asymptotic theory of the centered moments of the one-level density: Theorem 1.8 (taken from [Coh+26]) states that under GRH and a Fourier-support condition $\operatorname{supp}(\hat{\varphi}) \subset (-\sigma/n,\sigma/n)$ with $1\le \sigma\le 2$, the $n$-th centered moment over the family converges to $(n-1)!!\,(V_\varphi^2)^{n/2} \pm S(n,a;\varphi)$, where $S(n,a;\varphi)$ is an explicit iterated integral of $\hat{\varphi}$ and $\varphi$. Two test functions carry the argument. The first is the sign-changing function $\varphi_\omega(x) = (1-(x/\omega)^2)\,\hat{g}(x)$, built from an auxiliary even function $h$ by rescaling and convolution; it is nonnegative exactly on $[-\omega,\omega]$ and nonpositive outside, so its positive contribution to the one-level statistic can only come from zeros inside the window, and this sign pattern converts an odd moment inequality into a statement about how many forms have zeros there. The second is an auxiliary nonnegative Schwartz function (Lemma 3.1) that is bounded below on $[-\omega,\omega]$ and has Fourier support in $(-1/n,1/n)$; it makes the $2n$-th moment of the zero count $R_f(\omega)$ uniformly bounded. The final step is Cauchy–Schwarz: the moment gap $\Delta = S(n,a;\varphi_\omega)-(-\mu)^n > 0$ bounds the average of the nonnegative quantities $\Pi_f = (X_f-\mu)^n-(-\mu)^n$ from below, the second moment of $\Pi_f$ is bounded by the same constant $C$, and $a_{\omega,N}/|F_N| \ge \Delta^2/(4C)$ follows.

Extended reading notes

Core claim

The paper's central claim is a count rather than an existence: the number $a_{\omega,N}$ of even newforms of level $N$ whose $L$-function has at least one normalized zero in $(-\omega,\omega)$ obeys the lower bounds of Theorems 1.15 and 1.18. Theorem 1.15 says that if $\omega$ satisfies $-\bigl(\hat{\varphi}(0)+\tfrac12\int \hat{\varphi}(y)\,dy\bigr) < S(1,\sigma;\varphi_\omega)$, then along a sequence of prime levels $a_{\omega,N} \gg N^{7/8}\log N$. Theorem 1.18 says that if for some odd $n\ge 3$ the strict inequality $(-\mu)^n < S(n,a;\varphi_\omega)$ holds, with $\mu$ the limiting mean of the one-level density, then $a_{\omega,N}/|F_N|$ stays bounded away from zero: a positive proportion of the family carries a zero in the window. From this the paper derives that some exact integer $r\ge 1$ occurs as the number of zeros in the window on a positive proportion of forms, that forms with at least $r$ zeros have density at most $C_1/r^{2n}$, and that a positive proportion of forms have between one and $r_0-1$ zeros. The unifying mechanism is that the sign-changing test function $\varphi_\omega$ makes the odd centered-moment excess detectable only through forms with zeros inside the window, while a uniformly bounded even moment of the zero count $R_f(\omega)$ prevents that excess from being concentrated on a negligible set of forms.

Load-bearing premise

The load-bearing premise is that some odd moment order $n\ge 3$ and some window $\omega$ satisfy the strict inequality $(-\mu)^n < S(n,a;\varphi_\omega)$, where $\mu$ is the family's average one-level density and $S(n,a;\varphi_\omega)$ is a computable 'moment-excess' integral; the paper assumes this inequality holds without verifying it for any explicit $n$ or $\omega$ or supplying numerical evidence, so if it fails for every odd $n\ge 3$ the positive-proportion theorem has no instances (and GRH is assumed throughout).

Editorial extensions

If this is right

Under the first-moment condition, the number of forms with a zero in the window satisfies $a_{\omega,N} \gg N^{7/8}\log N$ infinitely often in prime level $N$, so the phenomenon already goes well beyond 'at least one form'.
For any window satisfying the odd-moment condition, a fixed positive fraction $\Delta^2/(4C)$ of the family has at least one zero in the window, making such zeros typical rather than exceptional.
The distribution of the number of zeros in the window has atoms: some positive integer $r$ occurs as the exact zero count on a positive proportion of forms (Corollary 1.20).
Forms with unusually many zeros in the window are polynomially rare: the density of forms with at least $r$ zeros is at most $C_1/r^{2n}$, and a positive proportion of forms carry between 1 and $r_0-1$ zeros (Corollaries 3.3 and 3.4).
Because $|F_N| \asymp N$ for fixed weight, the positive-proportion statement is a linear lower bound $a_{\omega,N} \gg_{\omega,n,k} N$ on the number of forms with a zero (Corollary 1.19).

Reading between the lines

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

If the moment-excess inequality $(-\mu)^n < S(n,a;\varphi_\omega)$ were verified numerically, say for the explicit choice $h(u)=\cos(\pi u/2)$ at $\omega\approx 0.21864$ and small odd $n$, Theorem 1.18 would immediately yield an explicit positive proportion; the paper itself does not perform that verification, so the constant is not yet explicit.
The same template—an odd moment excess detected by a sign-changing test function, converted by a bounded even moment of the zero count into a positive proportion—should transfer to other orthogonal families such as quadratic twists or Dirichlet $L$-functions wherever centered-moment asymptotics with an $S$-term are available, so the phenomenon is likely generic rather than special to prime-level n
Letting the window shrink slowly with the level ($\omega_N \to 0$) is the natural next test: if the moment excess survives in that regime, the methods could begin to say something about how many forms vanish exactly at the central point, connecting through modularity and the Birch–Swinnerton-Dyer conjecture to the distribution of analytic ranks in the family.

Original abstract

We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms. We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms. Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero. We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$. We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window. Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.64 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.15 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms. We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms. Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero. We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$. We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window. Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study low-lying zeros in families of even holomorphic cuspidal newforms of fixed weight and prime level, with particular emphasis on the number of forms having a zero in a prescribed normalized window about the central point and on the distribution of the number of such zeros among the forms.	contribution 0.91	0.90	0.09	asserted 0.81
s1	We quantify the number of forms having at least one zero in the window and study the distribution of the number of zeros in that window among the forms.	contribution 0.98	0.94	0.09	asserted 0.99
s2	Assuming the Generalized Riemann Hypothesis, we obtain new lower bounds for the number of forms having a low-lying zero.	result 0.80	0.93	0.22	asserted 0.99
s3	We first prove that, along an infinite sequence of prime levels $N$, the number of such forms is $\gg N^{7/8}\log N$.	result 0.99	0.95	0.22	asserted 1.00
s4	We then use higher centered moments and an appropriate test function to show that a positive proportion of the family has a zero in a prescribed window.	method 0.86	0.95	0.14	asserted 1.00
s5	Finally, we obtain polynomial upper-tail bounds for the number of zeros occurring there and show that a positive proportion of forms have a bounded, nonzero number of low-lying zeros.	result 0.98	0.96	0.34	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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P abstract

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References

Works cited (25)

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Palak Arora and Glenn Bruda and Bruce Fang and Raul Marquez and Steven J. Miller and Beni Prapashtica and Vismay Sharan and Daeyoung Son and Xueyiming Tang and Saad Waheed , title =. Journal of Number Theory , volume =. 2025 , pages =

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Palak Arora and Glenn Bruda and Bruce Fang and Raul Marquez and Steven J. Miller and Beni Prapashtica and Vismay Sharan and Daeyoung Son and Xueyiming Tang and Saad Waheed , title =. Journal of Number Theory , volume =. 2025 , pages =

View on Pith 2025

[2] [2]

Owen Barrett and Paula Burkhardt and Jonathan DeWitt and Robert Dorward and Steven J. Miller , title =. Research in Number Theory , volume =. 2017 , pages =

View on Pith 2017

[3] [3]

Mason Beckwith and Victor Chen and Matthew DeMora and Steven J. Miller and Anthony Zanello , title =. Journal of Number Theory , volume =. 2015 , pages =

View on Pith 2015

[4] [4]

Bryan J. Birch and H. Peter F. Swinnerton-Dyer , title =. Journal f\"ur die reine und angewandte Mathematik , volume =. 1963 , pages =

View on Pith 1963

[5] [5]

Bryan J. Birch and H. Peter F. Swinnerton-Dyer , title =. Journal f\"ur die reine und angewandte Mathematik , volume =. 1965 , pages =

View on Pith 1965

[6] [6]

Peter Cohen and Justine Dell and Oscar E. Gonz\'alez and Simran Khunger and Chung-Hang Kwan and Steven J. Miller and Alexander Shashkov and Alicia Smith Reina and Carsten Sprunger and Nicholas Triantafillou and Nhi Truong and Roger Van Peski and Stephen Willis , title =. Algebra & Number Theory , volume =. 2026 , pages =

View on Pith 2026

[7] [7]

Daniel Fiorilli and Steven J. Miller , title =. Algebra & Number Theory , volume =. 2015 , pages =

View on Pith 2015

[8] [8]

Christopher P. Hughes and Steven J. Miller , title =. Duke Mathematical Journal , volume =. 2007 , pages =

View on Pith 2007

[9] [9]

Christopher P. Hughes and Ze\'ev Rudnick , title =. The Quarterly Journal of Mathematics , volume =. 2003 , pages =

View on Pith 2003

[10] [10]

Henryk Iwaniec and Wenzhi Luo and Peter Sarnak , title =. Publications Math\'ematiques de l'IH\'ES , volume =. 2000 , pages =

View on Pith 2000

[11] [11]

Henryk Iwaniec and Emmanuel Kowalski , title =

View on Pith

[12] [12]

Nicholas M. Katz and Peter Sarnak , title =

View on Pith

[13] [13]

Nicholas M. Katz and Peter Sarnak , title =. Bulletin of the American Mathematical Society , volume =. 1999 , pages =

View on Pith 1999

[14] [14]

Stephen D. Miller , title =. Advances in Mathematics , volume =. 2006 , pages =

View on Pith 2006

[15] [15]

Jean-Fran. Formules explicites et minorations de conducteurs de vari\'et\'es alg\'ebriques , journal =. 1986 , pages =

View on Pith 1986

[16] [16]

Hugh L. Montgomery , title =. Analytic Number Theory, Proceedings of Symposia in Pure Mathematics , volume =. 1973 , pages =

View on Pith 1973

[17] [17]

Ze\'ev Rudnick and Peter Sarnak , title =. Duke Mathematical Journal , volume =. 1996 , pages =

View on Pith 1996

[18] [18]

Goes, John and Miller, Steven J. , title =. Journal of Number Theory , volume =. 2010 , doi =

View on Pith 2010

[19] [19]

Montgomery, Hugh L. , title =. Analytic number theory , year =

View on Pith

[20] [20]

Hejhal, Dennis A. , title =. International Mathematics Research Notices , year =

View on Pith

[21] [21]

Rudnick, Ze. Zeros of principal. Duke Mathematical Journal , year =

View on Pith

[22] [22]

Odlyzko, Andrew M. , title =. Mathematics of Computation , year =

View on Pith

[23] [23]

Hughes, C. P. , title =. Journal of Physics A: Mathematical and General , year =

View on Pith

[24] [24]

Bogomolny, E. and Bohigas, O. and Leboeuf, P. and others , title =. Journal of Statistical Physics , year =

View on Pith

[25] [25]

Conrey, J. Brian , title =. Mathematics unlimited—2001 and beyond , year =

View on Pith 2001