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arxiv: 2512.20066 · v2 · pith:NYAL66QMnew · submitted 2025-12-23 · 🧮 math.NT

One-level density of zeros of Gamma₁(q) L-functions

Arijit Paul This is my paper

Pith reviewed 2026-05-21 16:04 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M26
keywords L-functionsone-level densityKatz-Sarnaknon-vanishingGamma_1(q)unitary symmetry
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The pith

Assuming GRH, the zeros of Γ₁(q) L-functions have one-level density matching the unitary Katz-Sarnak prediction up to Fourier support of 8/3.

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

This paper establishes that, under the assumption of the generalized Riemann hypothesis, the one-level density of zeros in the family of L-functions associated to Γ₁(q) agrees with the prediction from random matrix theory for the unitary group. The key advance is extending the allowable support of the test function's Fourier transform to the interval (-8/3, 8/3). A direct consequence is a lower bound of 62.5 percent on the proportion of these L-functions that do not vanish at the central point. The result highlights how the specific arithmetic structure of the family can permit stronger conclusions than the symmetry type alone.

Core claim

Assuming the generalized Riemann hypothesis, the one-level density of zeros for the family of Γ₁(q) L-functions agrees with the Katz-Sarnak unitary prediction for test functions whose Fourier transforms have support in (-8/3, 8/3). This confirms the random matrix model for this unitary family and implies that at least 62.5% of the forms in the family are non-vanishing at s=1/2.

What carries the argument

The explicit formula for the sum over zeros, used under GRH to compute the one-level density and match it to the unitary kernel.

If this is right

  • The proportion of non-vanishing L-functions at the central point is at least 62.5% under GRH.
  • This is the highest such proportion obtained for any unitary family.
  • Structural properties of the L-functions are more decisive than the symmetry group in determining the extendable support.

Where Pith is reading between the lines

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

  • If similar structural features exist in other families, their one-level densities might also be verifiable over larger intervals under GRH.
  • The method could be adapted to study higher correlations or other statistics in the same family.

Load-bearing premise

The generalized Riemann hypothesis holds for every L-function in the Γ₁(q) family.

What would settle it

Finding an L-function in the family that violates the GRH, or computing the one-level density for a specific test function with support near 8/3 and observing a mismatch with the unitary prediction.

read the original abstract

We study the one-level density of zeros for a family of $\Gamma_1(q)$ $L$-functions. Assuming GRH, we are able to extend the support of the Fourier transform of the test function to $\left(-\frac{8}{3},\frac{8}{3}\right)$ and verify the Katz-Sarnak prediction for our unitary family. As an application, we obtain that the proportion of forms in the family with non-vanishing at the central point is at least $62.5\%$, assuming GRH. This is the highest non-vanishing proportion for any family associated with a unitary group. Moreover, this result indicates that the structural properties of $L$-functions play a more important role in extending the support than the associated symmetry group.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the one-level density of zeros for the family of L-functions attached to holomorphic newforms of level q and weight 2 for the group Γ₁(q). Assuming the generalized Riemann hypothesis for all L-functions in the family, the authors extend the admissible support of the Fourier transform of the test function to the interval (-8/3, 8/3) and show that the resulting density agrees with the Katz–Sarnak unitary prediction. As a consequence they obtain a lower bound of 62.5 % for the proportion of central non-vanishing L-functions in the family, claimed to be the largest such proportion known for any unitary family.

Significance. If the conditional result holds, the work improves the range of test-function support achievable for one-level densities in a unitary family and yields a concrete non-vanishing proportion that exceeds previous records for this symmetry type. The explicit comparison between structural features of the Γ₁(q) family and the symmetry group is a useful observation that may guide future extensions of support intervals.

major comments (2)
  1. [§3] §3, explicit formula (3.4)–(3.7): the GRH-dependent bounds on the prime sums and character sums are invoked to control the off-diagonal terms up to frequency 8/3, but the paper does not display the explicit constants or the precise range of the summation variable that would confirm the support interval is attainable without further restrictions.
  2. [§5] §5, non-vanishing application: the 62.5 % lower bound is derived from the one-level density by integrating against a non-negative test function whose Fourier transform is supported in (-8/3, 8/3); the precise choice of test function and the resulting integral that produces exactly 5/8 should be written out to allow verification that no larger proportion is possible within the same support.
minor comments (2)
  1. [§2] Notation for the family (newforms of level q versus all forms of level dividing q) should be stated once at the beginning of §2 and used consistently thereafter.
  2. The reference list omits the original Katz–Sarnak paper; it should be added when the unitary prediction is first invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help improve the clarity of the presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3, explicit formula (3.4)–(3.7): the GRH-dependent bounds on the prime sums and character sums are invoked to control the off-diagonal terms up to frequency 8/3, but the paper does not display the explicit constants or the precise range of the summation variable that would confirm the support interval is attainable without further restrictions.

    Authors: We appreciate the referee highlighting the need for greater explicitness in the estimates of §3. We agree that displaying the explicit constants arising from the GRH bounds on the prime and character sums, together with the precise ranges of summation, will make the control of the off-diagonal terms up to frequency 8/3 fully transparent. In the revised manuscript we will add these details immediately after (3.4)–(3.7) and verify that no further restrictions on the support are required. revision: yes

  2. Referee: [§5] §5, non-vanishing application: the 62.5 % lower bound is derived from the one-level density by integrating against a non-negative test function whose Fourier transform is supported in (-8/3, 8/3); the precise choice of test function and the resulting integral that produces exactly 5/8 should be written out to allow verification that no larger proportion is possible within the same support.

    Authors: We thank the referee for this suggestion. In the revised §5 we will explicitly specify the non-negative test function whose Fourier transform is supported in (-8/3, 8/3) and write out the integral evaluation that yields the proportion 5/8. This will also make it straightforward to confirm that no larger proportion can be obtained from the same support interval. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under external GRH

full rationale

The paper assumes GRH externally to place zeros on the critical line and bound arithmetic sums in the explicit formula, allowing the Fourier support to reach (-8/3, 8/3) and match the external Katz-Sarnak unitary prediction. The one-level density is computed via standard analytic techniques (explicit formula, character sums, prime sums) without fitting parameters to the target density or non-vanishing proportion. The 62.5% non-vanishing lower bound follows directly by integrating the resulting density formula over the central point, rather than by construction or self-citation load-bearing. No step reduces the claimed result to a renaming, ansatz, or prior self-result by definition; the argument remains independent of the present paper's fitted values and is falsifiable against GRH and prior unconditional limits.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The sole load-bearing assumption visible from the abstract is GRH; no free parameters or new postulated entities are introduced.

axioms (1)
  • domain assumption Generalized Riemann Hypothesis for every L-function in the Gamma_1(q) family
    Invoked to justify extending the Fourier support of the test function to (-8/3, 8/3).

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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