If our chaotic operator is derived correctly, then the Riemann hypothesis holds true
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914218052157440 |
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| author | Rafik, Zeraoulia Caceres, Pedro |
| author_facet | Rafik, Zeraoulia Caceres, Pedro |
| contents | This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis. Starting from the explicit zero-counting function $N(T)$, we construct a discrete map on the critical line and analyse its Lyapunov exponents and bifurcation diagrams, showing that the smooth von Mangoldt term generates a strongly unstable flow that captures the global growth of the zero density. Motivated by this dynamics, we define a self-adjoint ``chaotic'' operator $\mathcal{O}_α$ on a weighted Hilbert space with weight $\mathrm{d}N/\mathrm{d}T$, prove its unboundedness and essential self-adjointness, and describe its spectral resolution via the spectral theorem. Finite-dimensional truncations of $\mathcal{O}_α$ yield Hermitian random matrices whose eigenvalue statistics agree numerically with Gaussian unitary ensemble predictions and show qualitative similarities to both Odlyzko's zeta zeros and the hydrogen-atom spectrum, suggesting that $\mathcal{O}_α$ lies in the same universality class as the nontrivial zeros and providing a concrete Hilbert--Pólya--type framework rather than a proof of the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | If our chaotic operator is derived correctly, then the Riemann hypothesis holds true Rafik, Zeraoulia Caceres, Pedro General Mathematics Chaotic Dynamics 47Sxx-19F05-37P99-39A22-15A33 This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis. Starting from the explicit zero-counting function $N(T)$, we construct a discrete map on the critical line and analyse its Lyapunov exponents and bifurcation diagrams, showing that the smooth von Mangoldt term generates a strongly unstable flow that captures the global growth of the zero density. Motivated by this dynamics, we define a self-adjoint ``chaotic'' operator $\mathcal{O}_α$ on a weighted Hilbert space with weight $\mathrm{d}N/\mathrm{d}T$, prove its unboundedness and essential self-adjointness, and describe its spectral resolution via the spectral theorem. Finite-dimensional truncations of $\mathcal{O}_α$ yield Hermitian random matrices whose eigenvalue statistics agree numerically with Gaussian unitary ensemble predictions and show qualitative similarities to both Odlyzko's zeta zeros and the hydrogen-atom spectrum, suggesting that $\mathcal{O}_α$ lies in the same universality class as the nontrivial zeros and providing a concrete Hilbert--Pólya--type framework rather than a proof of the conjecture. |
| title | If our chaotic operator is derived correctly, then the Riemann hypothesis holds true |
| topic | General Mathematics Chaotic Dynamics 47Sxx-19F05-37P99-39A22-15A33 |
| url | https://arxiv.org/abs/2404.00583 |