Weil's quadratic form via the screw function

Masatoshi Suzuki

arxiv: 2606.09096 · v1 · pith:4NKCEJZAnew · submitted 2026-06-08 · 🧮 math.NT · math.FA

Weil's quadratic form via the screw function

Masatoshi Suzuki This is my paper

Pith reviewed 2026-06-27 15:05 UTC · model grok-4.3

classification 🧮 math.NT math.FA

keywords Weil quadratic formscrew functionRiemann zeta functionnontrivial zerosself-adjoint operatornonlocal differential operatorspectral interpretation

0 comments

The pith

The screw function recasts the Weil quadratic form using continuous functions and yields a conjecture for a limit operator with zeta zero eigenvalues.

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

The paper unifies results on the Weil quadratic form from Yoshida, Bombieri, and Connes-Consani by showing that the screw function supplies a representation in continuous functions rather than distributions. This representation is compatible with those earlier constructions. From the unified framework the author derives a conjecture that self-adjoint operators built from nonlocal realizations of the first-order differential operator on the interval [-a,a] converge, as a tends to infinity, to an operator whose eigenvalues are exactly the imaginary parts of the nontrivial zeros of the Riemann zeta function. The entire development avoids any assumption of the Riemann Hypothesis. The conjecture is presented as a spectral-theoretic counterpart to a related limit formula involving zeta-regularized products.

Core claim

The screw function provides a representation of the Weil quadratic form by continuous functions that is compatible with the constructions of Yoshida, Bombieri, and Connes-Consani. This representation leads to the conjecture that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function arises as the limit, as a tends to infinity, of self-adjoint operators coming from nonlocal realizations of the first-order differential operator on the finite interval [-a,a]. All statements hold without assuming the Riemann Hypothesis.

What carries the argument

The screw function, which supplies a continuous-function representation of the Weil quadratic form compatible with prior distributional constructions.

If this is right

The Weil quadratic form becomes accessible to direct study with ordinary continuous functions.
A concrete spectral operator whose spectrum encodes the zeta zeros can be obtained by taking the indicated limit.
The approach connects the Weil form to the limit formula for the zeta function expressed via zeta-regularized products.
The spectral interpretation of the nontrivial zeros receives an explicit operator-theoretic model that does not require the Riemann Hypothesis.

Where Pith is reading between the lines

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

Finite-a truncations might be used to approximate the imaginary parts of zeta zeros by solving eigenvalue problems on bounded intervals.
The same limit construction could be tested against other L-functions whose zeros are expected to obey analogous spectral laws.
Compatibility with the Connes-Consani-Moscovici limit formula suggests a possible dictionary between the screw-function representation and regularized-product expressions.

Load-bearing premise

The screw function supplies a representation of the Weil quadratic form by continuous functions that works with the earlier constructions without extra hidden assumptions on the distributions.

What would settle it

Numerically construct the finite-a operators for successively larger a, extract their eigenvalues, and check whether those eigenvalues converge to the known imaginary parts of the first several nontrivial zeta zeros.

read the original abstract

We establish a unified framework for understanding the results on the Weil quadratic form obtained by Yoshida (1992), Bombieri (2001, 2003), Connes--Consani (2023), and Connes--Consani--Moscovici (2025+) from the perspective of the screw function introduced in Suzuki (2023). An advantage of the approach via the screw function is that it provides a method to study the Weil quadratic form, which is originally defined in terms of distributions, by means of continuous functions. Based on this framework, we formulate a conjecture stating that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function can be obtained as the limit, as $a \to \infty$, of self-adjoint operators arising from nonlocal realizations of the first-order differential operator on the finite interval $[-a,a]$. All these results are obtained without assuming the Riemann Hypothesis. This conjecture may be compared with the limit formula for the Riemann zeta function expressed in terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and it sheds new light on the spectral-theoretic interpretation of the nontrivial zeros of the Riemann zeta function.

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

The paper states a concrete new conjecture on a limiting self-adjoint operator whose spectrum matches the imaginary parts of zeta zeros, but the unification via the screw function is asserted rather than shown in detail.

read the letter

The central new item is the conjecture that a self-adjoint operator with eigenvalues at the imaginary parts of the nontrivial zeta zeros arises as the a to infinity limit of operators coming from nonlocal realizations of the first-order differential operator on the interval [-a,a]. The paper also claims that the screw function from Suzuki's 2023 work supplies a continuous-function representation of the Weil quadratic form that brings together the approaches of Yoshida, Bombieri, and Connes-Consani.

This perspective does shift the Weil form from distributions to continuous functions, which could make some comparisons with trace formulas or regularized products easier to write down. The reorganization of the cited papers under one heading is the main organizational work.

The soft spot is that the conjecture is stated without any supporting calculation or limit argument in the text, and the unification step is not carried out explicitly here. Everything rests on the claim that the screw function reproduces the distributional pairings of the Weil form exactly. Without seeing the concrete conversion steps or checking boundary terms in the nonlocal realizations, it is not possible to tell whether hidden regularity conditions are needed or whether the distributional limits actually match. The dependence on the 2023 paper for the core representation adds a circularity burden that is not resolved in this manuscript.

This is a specialized note aimed at readers already following spectral interpretations of the zeta function and the Connes-Consani program. Someone working on explicit operators or limit formulas might find the conjecture worth testing against existing expressions, but the paper does not yet supply enough internal evidence to stand alone.

I would send it to referees because the conjecture is specific enough to be checked or compared with other models, even if substantial additional work is required to make the unification rigorous.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to establish a unified framework for the Weil quadratic form, drawing on results of Yoshida (1992), Bombieri (2001, 2003), Connes–Consani (2023) and Connes–Consani–Moscovici (2025+), by means of the screw function introduced in the author’s 2023 paper. The framework is said to replace the original distributional definition with an equivalent representation by continuous functions. On this basis the paper formulates a conjecture that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of ζ(s) arises as the a → ∞ limit of self-adjoint operators obtained from nonlocal realizations of the first-order differential operator on the finite interval [−a, a]. All statements are made without assuming the Riemann hypothesis.

Significance. If the claimed equivalence between the distributional Weil form and the continuous-function representation via the screw function can be made rigorous, the work would supply a concrete bridge between several existing approaches to the Weil quadratic form and a new spectral conjecture. The explicit avoidance of the Riemann hypothesis and the attempt to compare the conjecture with the Connes–Consani–Moscovici zeta-regularized product formula are positive features.

major comments (2)

[Abstract] Abstract and opening paragraphs of the main text: the central assertion that the screw function converts the distributional statements of Yoshida, Bombieri and Connes–Consani into statements about continuous functions is stated but the explicit conversion steps (change of test-function space, verification that pairings agree, control of support or regularity conditions) are not supplied. This conversion is load-bearing for both the unification claim and the subsequent conjecture.
[Framework section] Framework section (the part that invokes the 2023 screw-function definition): it is not shown that the nonlocal realizations of d/dx on [−a,a] produce boundary or cutoff terms whose distributional limits coincide exactly with the Weil quadratic form; any mismatch would invalidate the limit statement in the conjecture.

minor comments (1)

Notation for the screw function and its relation to the earlier 2023 definition should be made fully self-contained so that a reader need not consult the prior paper to follow the argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify places where the manuscript relies on prior work without sufficient self-contained detail. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract and opening paragraphs of the main text: the central assertion that the screw function converts the distributional statements of Yoshida, Bombieri and Connes–Consani into statements about continuous functions is stated but the explicit conversion steps (change of test-function space, verification that pairings agree, control of support or regularity conditions) are not supplied. This conversion is load-bearing for both the unification claim and the subsequent conjecture.

Authors: We agree that the explicit conversion steps are not supplied in sufficient detail in the current version. The 2023 screw-function paper supplies the underlying definitions, but the present manuscript does not reproduce the change-of-space argument, the verification that the pairings agree, or the support/regularity controls. In the revised manuscript we will insert a new subsection (immediately after the statement of the unification claim) that carries out these steps explicitly, citing the relevant propositions from Suzuki (2023) and verifying the necessary estimates on test functions. revision: yes
Referee: [Framework section] Framework section (the part that invokes the 2023 screw-function definition): it is not shown that the nonlocal realizations of d/dx on [−a,a] produce boundary or cutoff terms whose distributional limits coincide exactly with the Weil quadratic form; any mismatch would invalidate the limit statement in the conjecture.

Authors: The framework section invokes the screw-function representation to pass from the distributional Weil form to a continuous-function expression, and the conjecture is then stated in terms of the a → ∞ limit of the resulting operators. However, the manuscript does not contain a direct verification that the boundary and cutoff terms arising from the nonlocal realizations on [−a,a] have distributional limits that recover the Weil form exactly. This verification is indeed required for the limit statement to be well-posed. In the revision we will add an appendix that computes the relevant boundary terms, establishes the necessary convergence in the distributional sense, and confirms exact agreement with the Weil quadratic form. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework and conjecture are presented without internal reduction to inputs

full rationale

The paper references the screw function from the author's 2023 work to unify prior results on the Weil quadratic form and to formulate a conjecture on the limit of self-adjoint operators. No equations, derivations, or predictions are exhibited in the text that reduce by construction to quantities fixed by the present paper's own definitions or fits. The central output is a conjecture (not a theorem derived from the framework), and the self-citation supplies context rather than a load-bearing step that forces the result. This is a standard citation to prior independent work; the paper does not claim or show any self-definitional equivalence or fitted-input prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract alone; the full list of free parameters, axioms, and invented entities cannot be extracted. The construction appears to inherit the definition of the screw function and the properties of the Weil quadratic form from the cited literature.

pith-pipeline@v0.9.1-grok · 5731 in / 1261 out tokens · 26188 ms · 2026-06-27T15:05:52.442991+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages

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Yoshida, On Hermitian forms attached to zeta functions,Zeta functions in geometry (Tokyo, 1990), 281–325, Adv

H. Yoshida, On Hermitian forms attached to zeta functions,Zeta functions in geometry (Tokyo, 1990), 281–325, Adv. Stud. Pure Math., 21,Kinokuniya, Tokyo, 1992. Department of Mathematics, School of Science, Institute of Science Tokyo 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan Email:msuzuki@math.sci.isct.ac.jp

1990

[1] [1]

Bombieri, Remarks on Weil’s quadratic functional in the theory of prime numbers

E. Bombieri, Remarks on Weil’s quadratic functional in the theory of prime numbers. I,Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.11(2000), no. 3, 183–233 (2001)

2000

[2] [2]

Bombieri, A variational approach to the explicit formula,Comm

E. Bombieri, A variational approach to the explicit formula,Comm. Pure Appl. Math.56(2003), no. 8, 1151–1164

2003

[3] [3]

Connes, C

A. Connes, C. Consani, Spectral triples andζ-cycles,Enseign. Math.69(2023), no. 1-2, 93–148

2023

[4] [4]

Connes, C

A. Connes, C. Consani, H. Moscovici, Zeta spectral triples,https://arxiv.org/abs/2511.22755

work page arXiv

[5] [5]

Fukushima, Y

M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics,19,Walter de Gruyter & Co., Berlin, 2011

2011

[6] [6]

I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products. Seventh edition,Else- vier/Academic Press, Amsterdam, 2007

2007

[7] [7]

Kato, Perturbation theory for linear operators

T. Kato, Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in Mathe- matics,Springer-Verlag, Berlin, 1995

1980

[8] [8]

M. G. Kre˘ ın, H. Langer, Continuation of hermitian positive definite functions and related questions, Integral Equations Operator Theory78(2014), no. 1, 1–69

2014

[9] [9]

D. Lenz, P. Stollmann, I. Veseli´ c, The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms,Doc. Math.14(2009), 167–189

2009

[10] [10]

Olver, D

F. Olver, D. Lozier, R. Boisvert, C. Clark (eds), NIST handbook of mathematical functions,Cam- bridge University Press, Cambridge, 2010

2010

[11] [11]

M. Reed, B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975

1975

[12] [12]

Schm¨ udgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics 265,Springer, Dordrecht, 2012

K. Schm¨ udgen, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics 265,Springer, Dordrecht, 2012

2012

[13] [13]

Suzuki, Aspects of the screw function corresponding to the Riemann zeta function,J

M. Suzuki, Aspects of the screw function corresponding to the Riemann zeta function,J. Lond. Math. Soc.108(2023), no.4, 1448-1487

2023

[14] [14]

Suzuki, On the Hilbert space derived from the Weil distribution,Canadian Journal of Mathe- matics(FirstView)

M. Suzuki, On the Hilbert space derived from the Weil distribution,Canadian Journal of Mathe- matics(FirstView)

[15] [15]

formules explicites

A. Weil, Sur les “formules explicites” de la th´ eorie des nombres premiers,Comm. S´ em. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.],1952(1952), Tome Suppl´ ementaire, 252–265

1952

[16] [16]

Weil, Sur les formules explicites de la th´ eorie des nombres,Izv

A. Weil, Sur les formules explicites de la th´ eorie des nombres,Izv. Akad. Nauk SSSR Ser. Mat.,36 (1972), 3–18

1972

[17] [17]

Yoshida, On Hermitian forms attached to zeta functions,Zeta functions in geometry (Tokyo, 1990), 281–325, Adv

H. Yoshida, On Hermitian forms attached to zeta functions,Zeta functions in geometry (Tokyo, 1990), 281–325, Adv. Stud. Pure Math., 21,Kinokuniya, Tokyo, 1992. Department of Mathematics, School of Science, Institute of Science Tokyo 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan Email:msuzuki@math.sci.isct.ac.jp

1990