Construction of Finite Hilbert--Pólya Matrices from Weil's Explicit Formula
Abstract
Starting from the Riemann--Ξ specialization of Weil's explicit formula, we construct finite real-symmetric Prime--Weil matrices (S) from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert--Pólya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from N distinct positive ordinates γk, rational interpolation gives the exact contrast-pencil spectrum {±γ1,…,±γN} and positive-parity square spectrum {γ12,…,γN2}. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves λmin(S)→0. At N=L=13, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.
Related
- The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator
- A Combinatorial Formula and a Hilbert Scheme Realization of a Rank-One Virasoro Irregular Conformal Block
- Explicit formula for the Rasmussen invariant of odd pretzel knots
- Quantum Stokes matrices and quantum Riemann-Hilbert-Birkhoff maps
- On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm
BibTeX
@misc{shi2026construction,
title = {{Construction of Finite Hilbert--Pólya Matrices from Weil's Explicit Formula}},
author = {Yaoming Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04908},
eprint = {2609.04908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04908},
}
RIS
TY - UNPB
TI - Construction of Finite Hilbert--Pólya Matrices from Weil's Explicit Formula
AU - Yaoming Shi
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04908
UR - https://arxiv.org/abs/2609.04908
AB - Starting from the Riemann--Ξ specialization of Weil's explicit formula, we construct finite real-symmetric Prime--Weil matrices (S) from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert--Pólya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from N distinct positive ordinates γ_k, rational interpolation gives the exact contrast-pencil spectrum {±γ_1,…,±γ_N} and positive-parity square spectrum {γ_1^2,…,γ_N^2}. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves λ_min(S)→0. At N=L=13, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.
ER -