arxiv: 2605.12688 · v1 · submitted 2026-05-12 · 🧮 math.NT

Recognition: unknown

A connection between low-lying zeros and central values of L-functions

Didier Lesesvre , Ade Irma Suriajaya

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords L-functionslow-lying zeroscentral valuesKeating-Snaith conjectureRudnick-Sarnak density conjectureone-level densitysymmetry typesFourier support

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The pith

Partial results on low-lying zero distributions yield explicit conditional lower bounds on central values of L-functions.

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

The paper shows that classical proofs of the Rudnick-Sarnak density conjecture for low-lying zeros and the Keating-Snaith conjecture for central values share a crucial common ingredient. This overlap allows partial Fourier support results on the one-level density of zeros to be transferred into concrete lower bounds on the distribution of central values. The transfer works differently depending on the symmetry type of the L-function family and the width of the allowed Fourier support. A reader cares because it converts progress on zero statistics into measurable statements about how often central values are large, without needing the full conjectures.

Core claim

The same crucial ingredient appears in the classical approaches to both the Rudnick-Sarnak one-level density conjecture and the Keating-Snaith conjecture on central values; therefore partial results toward the former produce explicit conditional lower bounds toward the latter, with the quality of those bounds determined precisely by the symmetry type of the family and the size of the Fourier support permitted in the density statement.

What carries the argument

The shared crucial ingredient in the classical proofs of the Rudnick-Sarnak density conjecture and the Keating-Snaith conjecture, which transfers partial Fourier support results on zero distributions into lower bounds on value distributions.

If this is right

For orthogonal, unitary, or symplectic families, the allowed Fourier support directly controls the strength of the resulting lower bound on the proportion of large central values.
Explicit constants appear in the conditional lower bounds once the shared ingredient is isolated.
The relation between symmetry type, Fourier support interval, and bound quality is fully determined by the transfer argument.

Where Pith is reading between the lines

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

Similar transfers might apply to higher-level densities or other statistics of L-functions.
Numerical verification on specific families such as Dirichlet L-functions could test the predicted dependence on symmetry type.
The method suggests looking for analogous shared ingredients between other pairs of conjectures in analytic number theory.

Load-bearing premise

The assumption that the same key ingredient is needed in the standard approaches to both conjectures.

What would settle it

An explicit family of L-functions where a partial result on the one-level density of low-lying zeros fails to produce the corresponding lower bound on the distribution of central values.

read the original abstract

We discuss the relation between statistics on low-lying zeros of $L$-functions and distribution of the associated central values. More precisely, we deduce explicit conditional lower bounds toward the Keating-Snaith conjecture (on the distribution of central values of families of $L$-functions) from partial results toward the Rudnick-Sarnak density conjecture (on the one-level density for the low-lying zeros of these $L$-functions). We show in fact that the same crucial ingredient occurs in the classical approaches for proving both results, providing the connection. We precisely determine the relation between the type of symmetry of the family, the allowed Fourier support in its distributional statement, and the quality of the lower bounds obtained.

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

Partial results on low-lying zero densities transfer directly to explicit conditional lower bounds on central values via one shared analytic step.

read the letter

The main thing to know is that this paper shows how partial progress on the one-level density of low-lying zeros translates into explicit conditional lower bounds for the distribution of central values of L-functions, through a shared technical step in the classical methods for each. It does a good job making that transfer precise. The authors determine exactly how the symmetry type and the allowed Fourier support in the density result affect the quality of the central value bound you can get out. This is new and gives a clear dictionary rather than a vague link. The work is solid on its own terms because it works from existing partial results on Rudnick-Sarnak without introducing new assumptions or circular steps. The bounds are conditional, as expected, and the paper focuses on the mapping itself. A minor limitation is that the strength of the output bounds is capped by how much support has been achieved in the density side so far. For many families that is still limited, so the central value statements are not very strong yet. But the paper is upfront about working conditionally, so this is not a flaw. The citation pattern looks standard, referencing the relevant conjectures and prior partial results. This is for people in the field of L-function statistics who want to see how these two big conjectures are linked at the level of proofs. It will be most useful to those already comfortable with the explicit formula approaches and moment calculations involved. I would send it to peer review. The explicit correspondence is worth having in the literature and a referee can verify the details of the transfer.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to deduce explicit conditional lower bounds toward the Keating-Snaith conjecture on the distribution of central values of families of L-functions from partial results toward the Rudnick-Sarnak one-level density conjecture on low-lying zeros. It identifies a shared crucial analytic ingredient (an explicit formula or smoothed moment computation with restricted Fourier support) in the classical approaches to both, and determines a precise dictionary relating the symmetry type of the family, the admissible Fourier support size, and the resulting quality of the lower bounds obtained.

Significance. If the transfer via the shared ingredient is valid, the result would be significant for analytic number theory by providing a direct link between zero statistics and central value distributions, allowing partial progress on Rudnick-Sarnak to yield concrete conditional bounds on Keating-Snaith. The explicit dictionary and conditional nature make the connection falsifiable and potentially useful for further work on L-function families.

major comments (2)

[§2] §2 (or the section detailing the shared ingredient): The assertion that the same crucial ingredient occurs in both classical approaches requires an explicit side-by-side derivation or citation of the precise formula (e.g., the smoothed moment or explicit formula with Fourier support restriction) to confirm transferability without hidden assumptions or adjustments.
[§4] §4 (dictionary relating support and bounds): The explicit relation between symmetry type, admissible support size, and lower bound quality should include a concrete formula or table showing how the bound constant scales with the support parameter (e.g., for support up to 1 or 2), as the current description leaves the dependence implicit.

minor comments (2)

[Abstract] The abstract and introduction use 'we deduce' and 'we show in fact'; clarify whether the deduction is fully self-contained or relies on citing specific partial results from prior works on Rudnick-Sarnak.
Notation for the Fourier support restriction and symmetry types (e.g., orthogonal, unitary) should be standardized across sections to avoid ambiguity when stating the dictionary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The suggested clarifications will strengthen the presentation of the shared analytic ingredient and the explicit dictionary. We address the major comments below.

read point-by-point responses

Referee: [§2] §2 (or the section detailing the shared ingredient): The assertion that the same crucial ingredient occurs in both classical approaches requires an explicit side-by-side derivation or citation of the precise formula (e.g., the smoothed moment or explicit formula with Fourier support restriction) to confirm transferability without hidden assumptions or adjustments.

Authors: We agree that an explicit side-by-side comparison is needed to confirm the transfer without hidden adjustments. In the revised manuscript we will expand the relevant section (currently §2) to include a direct comparison: we will recall the explicit formula for the one-level density (with the indicated Fourier support restriction) from the Rudnick–Sarnak approach and the corresponding smoothed moment expression from the Keating–Snaith approach, placing the two formulas side by side and citing the original derivations (Rudnick–Sarnak 1996 and Keating–Snaith 2000) for each step. revision: yes
Referee: [§4] §4 (dictionary relating support and bounds): The explicit relation between symmetry type, admissible support size, and lower bound quality should include a concrete formula or table showing how the bound constant scales with the support parameter (e.g., for support up to 1 or 2), as the current description leaves the dependence implicit.

Authors: We thank the referee for this observation. Although §4 already determines the precise relation between symmetry type, admissible Fourier support, and the resulting lower bound, we will make the scaling fully explicit by adding a table that lists, for each symmetry type (unitary, symplectic, orthogonal), the maximal admissible support interval and the explicit constant appearing in the conditional lower bound for the proportion of large central values. The table will include the cases of support contained in [-1,1] and [-2,2] and will state the functional dependence of the bound constant on the support length. revision: yes

Circularity Check

0 steps flagged

No significant circularity; conditional transfer between distinct conjectures

full rationale

The paper deduces explicit conditional lower bounds toward the Keating-Snaith conjecture from partial results on the Rudnick-Sarnak one-level density conjecture by exhibiting a shared analytic ingredient (explicit formula or smoothed moment with restricted Fourier support) that appears in the classical approaches to both. It then supplies an explicit dictionary relating symmetry type, admissible support size, and resulting bound quality. This is a direct transfer of independent external partial results rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or step reduces the claimed output to the input by construction; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions in analytic number theory about L-function families and their zero statistics; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption Partial results toward the Rudnick-Sarnak one-level density conjecture hold for the families under consideration
This is the input used to obtain the lower bounds on central values.
ad hoc to paper The classical approaches to both conjectures share a crucial common ingredient
This shared ingredient is the mechanism that transfers the zero-density information to central-value bounds.

pith-pipeline@v0.9.0 · 5413 in / 1312 out tokens · 47725 ms · 2026-05-14T19:41:02.573632+00:00 · methodology

discussion (0)

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Works this paper leans on

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