The Riemann Hypothesis: A Hilbert–Pólya Candidate Operator.

A Hilbert–Pólya Candidate for the Riemann Zeros:

We present a finite-dimensional, self-adjoint operator that numerically reproduces the principal analytic and statistical properties conjectured for a Hilbert–Pólya operator whose eigenvalues would correspond to the nontrivial zeros of the Riemann zeta function.

The operator is constructed as a sum of three components: an arithmetic diagonal encoding the Riemann–von Mangoldt density, a resonance-tuned SECH-squared kernel weighted by the von Mangoldt function, and a low-rank, prime-modulated kernel that injects explicit prime oscillations into the spectrum. A small Gaussian perturbation is added to achieve chaotic level statistics. The operator is then embedded into a block form that enforces exact spectral reflection symmetry and eigenvector orthogonality at machine precision.

Extensive numerical validation across dimensions up to two thousand establishes the following finite-N results:

• Proposition 1 (Self-adjointness & Real Spectrum). The operator is exactly self-adjoint; its eigenvalues are real.
• Numerical Observation 2 (Weyl Law). The eigenvalue counting function matches the Riemann–von Mangoldt asymptotic density within a relative error below one percent for the tested dimensions.
• Proposition 3 (Functional-Equation Symmetry). The block operator satisfies λ ↔ −λ pairing, equivalent to the functional equation of the zeta function, with normalized errors below 10^{-14}.
• Numerical Observation 4 (Explicit-Formula Trace Identity – Smoothed). For Gaussian test functions, the spectral trace is consistent with the prime-power side of the explicit formula up to a controlled truncation error.
• Numerical Observation 5 (GUE-Plus-Arithmetic Statistics). After Berry–Keating unfolding, the eigenvalue spacings exhibit statistics intermediate between Poisson and GUE, with the mean spacing ratio in the range ≈0.43–0.45, while the empirical distribution shows improving alignment with Riemann zero data as dimension increases.
• Numerical Observation 6 (Resolvent Convergence). Finite-N resolvent differences shrink as dimension increases, supporting heuristically the existence of a well-defined infinite-dimensional limit operator.

This construction provides one of the most comprehensive numerical explorations to date of a concrete Hilbert–Pólya candidate. It satisfies key finite-dimensional analytic properties and reproduces several important statistical features expected from the Riemann zeros. While these results constitute strong numerical evidence for the viability of the approach, they do not constitute a proof of the Riemann Hypothesis. It offers a concrete, testable model that can be further analyzed toward the global and local requirements conjectured by Hilbert and Pólya.