The Hecke Operator as D'Alembertian of the Zeta Pseudo-Riemannian Metric: Quaternionic Extension, Spectral Identification, and Hawking Radiation as the Prime Readout
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| Natura: | Recurso digital |
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2026
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| _version_ | 1866901448580661248 |
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| author | de Oliveira, Leandro |
| author_facet | de Oliveira, Leandro |
| contents | <p class="MsoNormal">We work backwards: starting from the known spectral structure of the Hecke operators on the adèlic space AQ and deriving the d'Alembertian □g of the zeta pseudo-Riemannian metric [1] as its restriction to the critical strip. The Hecke operator Tp acts on L 2 (AQ/Q<span>∗</span> ) with eigenvalues <span>λ</span>p(<span>ρ</span>n) = 2p <span>−</span>1/2 cos(tn log p) at each non-trivial zero <span>ρ</span>n = 1 2 +itn of <span>ζ</span>. We show that <span>□</span>g = g <span>σσ</span>(<span>∂</span> 2 <span>σ</span> <span>−∂</span> 2 t ) on the critical strip is the Archimedean restriction of this Hecke action, completing the local-global picture. The quaternionic extension Q2[i] = Q2<span>⊗</span>QQ(i) (the 2- adic Gaussian integers, base 4 over binary) provides the natural ambient space: the zeta metric extends from C to H via <span></span>(s, w) = <span>−</span> log |<span>ζ</span>(s)| <span>−</span> log |<span>ζ</span>(w)| for s, w conjugate quaternionic pairs, preserving the Lorentzian signature and the geodesic structure. The analogue Hawking radiation at the event horizon <span>σ</span> = 1 is identified with the explicit formula of Riemann: each zero <span>ρ</span>n emits an oscillatory contribution x <span>ρ</span>n /<span>ρ</span>n to <span>ψ</span>(x), precisely as a thermal quantum radiates energy from a black hole horizon. The null constant t<sup> </sup><sup><span>∗</span></sup> <span>≈</span> 5.5612 acquires ad<span>è</span>lic frequencies t <sup><span>∗</span></sup> <span>·</span>log p for each prime p, providing a new connection between t<sup> </sup><sup><span>∗</span></sup> and the Euler product structure of <span>ζ</span>.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19653775 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Hecke Operator as D'Alembertian of the Zeta Pseudo-Riemannian Metric: Quaternionic Extension, Spectral Identification, and Hawking Radiation as the Prime Readout de Oliveira, Leandro Hecke operator d'Alembertian pseudo-Riemannian metric quaternions adèlic space Tate's thesis Hawking radiation explicit formula spectral identification <p class="MsoNormal">We work backwards: starting from the known spectral structure of the Hecke operators on the adèlic space AQ and deriving the d'Alembertian □g of the zeta pseudo-Riemannian metric [1] as its restriction to the critical strip. The Hecke operator Tp acts on L 2 (AQ/Q<span>∗</span> ) with eigenvalues <span>λ</span>p(<span>ρ</span>n) = 2p <span>−</span>1/2 cos(tn log p) at each non-trivial zero <span>ρ</span>n = 1 2 +itn of <span>ζ</span>. We show that <span>□</span>g = g <span>σσ</span>(<span>∂</span> 2 <span>σ</span> <span>−∂</span> 2 t ) on the critical strip is the Archimedean restriction of this Hecke action, completing the local-global picture. The quaternionic extension Q2[i] = Q2<span>⊗</span>QQ(i) (the 2- adic Gaussian integers, base 4 over binary) provides the natural ambient space: the zeta metric extends from C to H via <span></span>(s, w) = <span>−</span> log |<span>ζ</span>(s)| <span>−</span> log |<span>ζ</span>(w)| for s, w conjugate quaternionic pairs, preserving the Lorentzian signature and the geodesic structure. The analogue Hawking radiation at the event horizon <span>σ</span> = 1 is identified with the explicit formula of Riemann: each zero <span>ρ</span>n emits an oscillatory contribution x <span>ρ</span>n /<span>ρ</span>n to <span>ψ</span>(x), precisely as a thermal quantum radiates energy from a black hole horizon. The null constant t<sup> </sup><sup><span>∗</span></sup> <span>≈</span> 5.5612 acquires ad<span>è</span>lic frequencies t <sup><span>∗</span></sup> <span>·</span>log p for each prime p, providing a new connection between t<sup> </sup><sup><span>∗</span></sup> and the Euler product structure of <span>ζ</span>.</p> |
| title | The Hecke Operator as D'Alembertian of the Zeta Pseudo-Riemannian Metric: Quaternionic Extension, Spectral Identification, and Hawking Radiation as the Prime Readout |
| topic | Hecke operator d'Alembertian pseudo-Riemannian metric quaternions adèlic space Tate's thesis Hawking radiation explicit formula spectral identification |
| url | https://doi.org/10.5281/zenodo.19653775 |