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March 27, 20260 citationsOpen Access

The Hilbert-Pólya Operator: A Candidate Construction via Redheffer-Mangoldt Symmetrization (v3 with Extended Tests and Honest Assessment)

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MBMichael Bieg

Key Points

  • The research aims to evaluate the Hilbert-Pólya operator's performance and its implications on spectral properties.
  • Conducted extended numerical tests up to n=5000.
  • Applied the Redheffer matrix with logarithmic weights.
  • Assessed spectral density and essential self-adjointness.
  • Non-zeta-zero points approximated equally by the spectrum.
  • Provided positive evidence for zeta zero approximation and log(p) eigenvalues.
  • Identified issues with non-monotone convergence and unproven essential self-adjointness.

Abstract

Version 3: Extended numerical tests up to n=5000 and honest reassessment. The operator Hₙ = (Rₙ Dₙ + Dₙ RₙT) /2 combines the Redheffer matrix with logarithmic weights. Key correction: extended tests reveal that non-zeta-zero points are approximated equally well by the spectrum (spectral density problem), undermining the spectral exclusivity claim from v2. The paper now presents both positive evidence (zeta zero approximation, log (p) eigenvalues, explicit formula connection) and negative evidence (spectral density, non-monotone convergence, unproven essential self-adjointness). Includes the extended test script for reproducibility. Previous 'theorems' (Attractor, Essential Self-Adjointness) downgraded to conjectures.

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Cite This Study

Michael Bieg (2026) studied this question.

synapsesocial.com/papers/69c620d515a0a509bde197e4https://doi.org/10.5281/zenodo.19218449
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