Goldilocks Completion: Hilbert-Polya Conjecture
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| Format: | Recurso digital |
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2025
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| _version_ | 1866901731099541504 |
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| author | Rey, Michael |
| author_facet | Rey, Michael |
| contents | <p>We construct, in a completely explicit and elementary way, a self-adjoint operator whose spectrum is given by the prime-local zero curves that arise in the Goldilocks analytic continuation of the Euler product for the Riemann zeta function. The construction uses only the prime number level, the explicit zero formula for the Goldilocks prime blocks, and basic Hilbert space theory. We work on the critical line Re(s) = 1/2, so that the zeros we use have the form s = 1/2 + it, and we take the real numbers t as eigenvalues of a diagonal operator on a log-prime orthonormal basis. We use the Weierstrass product with a simple real-reflection adjustment to ensure that the completed Goldilocks function is real on the critical line; this allows us to apply the functional calculus for self-adjoint operators in a completely concrete way, aligned with the Hilbert-P´olya conjecture for the classical nontrivial zeros of ζ(s).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17845050 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Goldilocks Completion: Hilbert-Polya Conjecture Rey, Michael <p>We construct, in a completely explicit and elementary way, a self-adjoint operator whose spectrum is given by the prime-local zero curves that arise in the Goldilocks analytic continuation of the Euler product for the Riemann zeta function. The construction uses only the prime number level, the explicit zero formula for the Goldilocks prime blocks, and basic Hilbert space theory. We work on the critical line Re(s) = 1/2, so that the zeros we use have the form s = 1/2 + it, and we take the real numbers t as eigenvalues of a diagonal operator on a log-prime orthonormal basis. We use the Weierstrass product with a simple real-reflection adjustment to ensure that the completed Goldilocks function is real on the critical line; this allows us to apply the functional calculus for self-adjoint operators in a completely concrete way, aligned with the Hilbert-P´olya conjecture for the classical nontrivial zeros of ζ(s).</p> |
| title | Goldilocks Completion: Hilbert-Polya Conjecture |
| url | https://doi.org/10.5281/zenodo.17845050 |