Spectral Operator Approach to the Riemann Hypothesis: Non-Hermitian Eigenvalues and the Critical Line — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Riemann Hypothesis (RH) remains unproven; computational evidence is extensive but non-exhaustive. A notable theoretical strategy links RH to eigenfunctions of a non-Hermitian operator, suggesting a spectral interpretation of the zeta zeros. | MATH: Riemann zeta function ζ(s) = Σ n⁻ˢ (Re(s)>1), analytically continued. RH: all non-trivial zeros satisfy Re(s) = 1/2 (critical line). The operator D⁺ (non-Hermitian) has eigenvalues = zeros of ζ(s); RH ⇔ orthogonality condition for its eigenfunctions. No closed-form ratio or new constant emerges from the cited sources. | CONNECTION: The critical line Re(s)=1/2 is a symmetry axis — a reflection symmetry in the complex plane. The spectral interpretation (Hilbert–Pólya) hints at a hidden self-adjoint operator, whose eigenvalues would be real (1/2 + iγ), implying a unitary symmetry — but no explicit geometric ratio (0.382, 0.618, etc.) or crystallographic group is directly evidenced. The base-60 connection is absent. | DEPTH: 8 — The
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026spectral,
title = {{Spectral Operator Approach to the Riemann Hypothesis: Non-Hermitian Eigenvalues and the Critical Line — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553955},
url = {https://zenodo.org/records/23553955},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Operator Approach to the Riemann Hypothesis: Non-Hermitian Eigenvalues and the Critical Line — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553955
UR - https://zenodo.org/records/23553955
AB - FINDING: The Riemann Hypothesis (RH) remains unproven; computational evidence is extensive but non-exhaustive. A notable theoretical strategy links RH to eigenfunctions of a non-Hermitian operator, suggesting a spectral interpretation of the zeta zeros. | MATH: Riemann zeta function ζ(s) = Σ n⁻ˢ (Re(s)>1), analytically continued. RH: all non-trivial zeros satisfy Re(s) = 1/2 (critical line). The operator D⁺ (non-Hermitian) has eigenvalues = zeros of ζ(s); RH ⇔ orthogonality condition for its eigenfunctions. No closed-form ratio or new constant emerges from the cited sources. | CONNECTION: The critical line Re(s)=1/2 is a symmetry axis — a reflection symmetry in the complex plane. The spectral interpretation (Hilbert–Pólya) hints at a hidden self-adjoint operator, whose eigenvalues would be real (1/2 + iγ), implying a unitary symmetry — but no explicit geometric ratio (0.382, 0.618, etc.) or crystallographic group is directly evidenced. The base-60 connection is absent. | DEPTH: 8 — The Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -