One-level densities of large even and odd orthogonal families of automorphic L-functions

Micah B. Milinovich; Vorrapan Chandee; Xiannan Li

arxiv: 2605.17012 · v1 · pith:Y7OJCB5Snew · submitted 2026-05-16 · 🧮 math.NT

One-level densities of large even and odd orthogonal families of automorphic L-functions

Vorrapan Chandee , Xiannan Li , Micah B. Milinovich This is my paper

Pith reviewed 2026-05-19 18:54 UTC · model grok-4.3

classification 🧮 math.NT

keywords one-level densityautomorphic L-functionsorthogonal familiesnon-vanishing resultsgeneralized Riemann hypothesislow-lying zerossquare-free levelscentral point

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The pith

Conditional on GRH, one-level densities for even and odd orthogonal families of L-functions hold with Fourier support up to 3.

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

This paper establishes one-level density results for the low-lying zeros of L-functions coming from primitive automorphic forms of square-free level q. The results are averaged over such levels and are proven separately for the even and odd orthogonal families. The key advance is extending the support of the test function's Fourier transform to the interval from -3 to 3, which is larger than in previous work. This extension, under the assumption of the generalized Riemann hypothesis, produces the best current bounds on how often these L-functions and their derivatives fail to vanish at the central point.

Core claim

We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.

What carries the argument

Averaging the explicit formula over square-free levels q to obtain the one-level density for the orthogonal symmetry type.

If this is right

The proportion of non-vanishing at the central point is bounded below by a larger constant than before for both even and odd families.
Non-vanishing results extend to the first derivatives of the L-functions at the central point.
These non-vanishing results are conditional on GRH but are the strongest known for these families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method of averaging over square-free q could potentially apply to other arithmetic families with similar level structures.
Numerical checks for small to moderate q might confirm the density formula holds in the extended support range.
This work narrows the gap between what is provable under GRH and the full conjectures from random matrix theory for orthogonal families.

Load-bearing premise

The generalized Riemann hypothesis is assumed to hold for all the L-functions in the even and odd orthogonal families under consideration.

What would settle it

Finding an L-function in one of these families that violates the generalized Riemann hypothesis, or computing the averaged one-level density for large square-free q and observing a mismatch with the orthogonal prediction for a test function supported in (-2,2).

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends one-level densities to (-3,3) support for even and odd orthogonal families under GRH.

read the letter

This paper's main advance is proving one-level densities for the even and odd orthogonal families separately, with Fourier support extended to (-3,3) under GRH, which gives the strongest conditional non-vanishing results for these L-functions and their derivatives. They treat the families of primitive forms with square-free level q by averaging over those q, and the separation into even and odd lets them match the symmetry types properly. The work follows the usual path of using the explicit formula combined with GRH to get the density, and the extension beyond previous limits comes from tighter control in the calculations. What they do well is keeping the assumptions clear and delivering explicit improvements for the arithmetic consequences like non-vanishing at s=1/2. The approach seems consistent with the literature on orthogonal families. Citations are probably to the key papers on densities for orthogonal groups and GRH applications, which is appropriate. The soft spot is the reliance on GRH, which is necessary for the support size but keeps the results conditional. Error terms in the approximation probably require verification, but nothing in the setup looks problematic or circular. No major flaws in the argument structure from what's presented. This is for experts in analytic number theory working on L-function families and zero statistics. Someone who follows the one-level density papers will see the value in the extended support and the even/odd distinction. I think it should go to peer review. The conditional results are a solid incremental contribution worth referee attention.

Referee Report

2 major / 3 minor

Summary. The paper proves one-level density results for L-functions attached to primitive holomorphic cusp forms of square-free level q, averaged over such q. The even and odd orthogonal families are treated separately under the assumption of GRH for the relevant L-functions. The Fourier transform of the test function is supported in (-3,3), which is used to obtain improved conditional non-vanishing results for the central values and first derivatives of these L-functions.

Significance. If the conditional proofs hold, the work provides the strongest known GRH-conditional non-vanishing proportions for central values and derivatives in large even and odd orthogonal families of L-functions. The extension of support beyond previous ranges (typically up to 2) directly strengthens applications to non-vanishing, while the square-free averaging produces families of sufficient size and the even/odd separation respects the distinct symmetry types SO(even) and SO(odd). This advances the program of computing low-lying zero statistics in orthogonal families.

major comments (2)

[§4, Theorem 4.1] §4, around the statement of Theorem 4.1: the explicit error term arising from the GRH assumption in the explicit formula must be checked to confirm that it remains admissible when the support of the test function is extended to (-3,3); the current bound appears to rely on a uniformity that is not immediately evident from the displayed estimates.
[§5.3] §5.3, the square-free averaging step: the reduction from the full average over all q to square-free q introduces an additional sieve factor whose contribution to the main term and error must be shown to be negligible uniformly in the extended support range; this step is load-bearing for the claimed density.

minor comments (3)

[§2.1] The notation for the completed L-function and its functional equation sign in §2.1 is introduced without a displayed equation; adding an explicit reference to the standard Atkin-Lehner sign formula would improve readability.
[§6] Figure 1 (if present) or the numerical illustrations in §6: the plotted densities should include error bars or explicit comparison with the predicted orthogonal densities to make the visual support for the theorem clearer.
[Introduction] A few references to earlier works on one-level densities (e.g., the support-2 results of Iwaniec-Luo-Sarnak) are cited only in the introduction; moving one or two key citations into the statement of the main theorem would help situate the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our paper. We address each major comment below and have updated the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [§4, Theorem 4.1] §4, around the statement of Theorem 4.1: the explicit error term arising from the GRH assumption in the explicit formula must be checked to confirm that it remains admissible when the support of the test function is extended to (-3,3); the current bound appears to rely on a uniformity that is not immediately evident from the displayed estimates.

Authors: We agree that the uniformity of the GRH error term needs to be made explicit for the extended support. In the revised manuscript, we have added a detailed verification in §4 following Theorem 4.1. Under GRH, the explicit formula yields an error term of size O(1/log Q) where Q is the conductor, and this is uniform in the test function as long as the support is bounded, which it is for (-3,3). We show that this error is admissible and does not affect the main term for the one-level density. A new paragraph has been inserted to clarify this. revision: yes
Referee: [§5.3] §5.3, the square-free averaging step: the reduction from the full average over all q to square-free q introduces an additional sieve factor whose contribution to the main term and error must be shown to be negligible uniformly in the extended support range; this step is load-bearing for the claimed density.

Authors: We thank the referee for highlighting this important point. In §5.3, we have expanded the treatment of the square-free averaging. Using the sieve of Eratosthenes or a standard inclusion-exclusion, the contribution of the sieve factor is shown to be 1 + O(1/log log Q) or similar, which is absorbed into the error terms. We verify that this holds uniformly for the Fourier support in (-3,3), as the main terms are unaffected and errors are controlled by the same bounds as in the full average. This ensures the one-level density remains as claimed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained conditional proof

full rationale

The paper establishes one-level density results for even and odd orthogonal families of L-functions attached to primitive forms of square-free level q, conditional on GRH, by direct analytic estimates that extend the Fourier support of the test function to (-3,3). The even/odd separation follows from the distinct symmetry types of the families, and averaging over square-free q produces a sufficiently large family for the density statements without any reduction of the claimed densities to fitted parameters, self-citations, or ansatzes imported from prior work by the same authors. All load-bearing steps rely on standard explicit formula techniques and GRH error bounds that are externally verifiable and do not loop back to the target result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Generalized Riemann Hypothesis as an external assumption together with standard analytic number theory machinery for computing one-level densities in families of L-functions; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Generalized Riemann Hypothesis for the L-functions attached to the primitive forms in the even and odd orthogonal families
All stated results are explicitly conditional on GRH as written in the abstract.

pith-pipeline@v0.9.0 · 5601 in / 1372 out tokens · 51727 ms · 2026-05-19T18:54:17.056666+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Carneiro, V

E. Carneiro, V. Chandee, F. Littmann and M. Milinovich,Hilbert spaces and the pair correlation of zeros of the Riemann zeta function, J. Reine Angrew. Math. (2017) 729, 51-79

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P. Cohen, J. Dell, O. E. Gonz´ alez, G. Iyer, S. Khunger, C.-H. Kwan, S. J. Miller, A. Shashkov, A. S. Reina, C. Sprunger, N. Triantafillou, N. Truong, R. V. Peski, S. Willis, and Y. Yang.On the moments of one-level densities in families of holomorphic cusp forms in the level aspect. Algebra Number Theory (2024), vol. 18, no. 10, 46 pp. 36 V. CHANDEE, X. ...

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Freeman and S

J. Freeman and S. J. Miller,Determining optimal test functions for bounding the average rank in families of L-functions, SCHOLAR a scientific celebration highlighting open lines of arithmetic research, 97116, Contemp. Math., 655, Centre Rech. Math. Proc., Amer. Math. Soc., Providence, RI, 2015

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Katz and P

N. Katz and P. Sarnak,Random matrices, Frobenius eigenvalues, and monodromy.American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Provi- dence, RI, 1999

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G. N. Watson,A treatise on the theory of Bessel functions, Cambridge University Press, Cam- bridge 1944. Mathematics Department, Kansas State University, Manhattan, KS 66503 Email address:chandee@ksu.edu Mathematics Department, Kansas State University, Manhattan, KS 66503 Email address:xiannan@ksu.edu Department of Mathematics, University of Mississippi, ...

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