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2025
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| Online Access: | https://doi.org/10.5281/zenodo.15259165 |
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| _version_ | 1866902143649185792 |
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| author | Washburn, Jonathan |
| author_facet | Washburn, Jonathan |
| contents | <p>We construct an explicit differential operator H = -ix∂x - i/2 + V(x) on L2(R+,x-1dx) whose potential V(x) = 2(φ/π)x + Vres(x) derives from the Recognition Theory cost functional. A full Sturm-Liouville analysis shows that H is essentially self-adjoint: both endpoints are limit-point even after inclusion of the prime-resonance series Vres. Diagonalising H with a golden-ratio Mellin transform produces an exact quantisation condition via the Weyl m-function, replacing earlier semiclassical estimates by a rigorous θ(eT) = nπ criterion. We then compute the Gelfand-Yaglom regularised determinant and prove det H - iφ(s-1/2) = ξ(s), the completed Riemann zeta function. An explicit bound |NH(T)-Nζ(T)| < 1/2 for all T establishes a one-to-one correspondence between the spectra, so every non-trivial zero lies on the critical line. Hence the Riemann Hypothesis holds as a corollary of spectral theory for a parameter-free operator. Potential extensions to L-functions and applications to prime-gap bounds are outlined.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15259165 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Self-Adjoint Recognition Operator with Discrete Spectrum Matching the Non-Trivial Zeros of the Riemann Zeta Function Washburn, Jonathan <p>We construct an explicit differential operator H = -ix∂x - i/2 + V(x) on L2(R+,x-1dx) whose potential V(x) = 2(φ/π)x + Vres(x) derives from the Recognition Theory cost functional. A full Sturm-Liouville analysis shows that H is essentially self-adjoint: both endpoints are limit-point even after inclusion of the prime-resonance series Vres. Diagonalising H with a golden-ratio Mellin transform produces an exact quantisation condition via the Weyl m-function, replacing earlier semiclassical estimates by a rigorous θ(eT) = nπ criterion. We then compute the Gelfand-Yaglom regularised determinant and prove det H - iφ(s-1/2) = ξ(s), the completed Riemann zeta function. An explicit bound |NH(T)-Nζ(T)| < 1/2 for all T establishes a one-to-one correspondence between the spectra, so every non-trivial zero lies on the critical line. Hence the Riemann Hypothesis holds as a corollary of spectral theory for a parameter-free operator. Potential extensions to L-functions and applications to prime-gap bounds are outlined.</p> |
| title | A Self-Adjoint Recognition Operator with Discrete Spectrum Matching the Non-Trivial Zeros of the Riemann Zeta Function |
| url | https://doi.org/10.5281/zenodo.15259165 |