Weil's quadratic form via the screw function
Abstract
We establish a unified operator-theoretic 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 and brings the theory of de Branges spaces into the analysis of the Weil quadratic form via the Fourier transform. 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 → ∞, 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.
Related
- Spectral convergence associated with Weil's quadratic form
- A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion
- A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form
- High-Precision Approximation of Riemann Zeros via the Truncated Weil Form
- A power-saving asymptotic for a Möbius-weighted GCD quadratic form
BibTeX
@misc{suzuki2026weils,
title = {{Weil's quadratic form via the screw function}},
author = {Masatoshi Suzuki},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.09096},
eprint = {2606.09096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09096},
}
RIS
TY - UNPB
TI - Weil's quadratic form via the screw function
AU - Masatoshi Suzuki
PY - 2026
DA - 2026/06/08
PB - arXiv
DO - 10.48550/arxiv.2606.09096
UR - https://arxiv.org/abs/2606.09096
AB - We establish a unified operator-theoretic 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 and brings the theory of de Branges spaces into the analysis of the Weil quadratic form via the Fourier transform. 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 → ∞, 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.
SN - 2331-8422
ER -