Complete Proof of the Riemann Hypothesis via Spectral Theory
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2027
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| _version_ | 1866902177906163712 |
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| author | Rodrigues, Vinicius Ramos |
| author_facet | Rodrigues, Vinicius Ramos |
| contents | <p>We present a rigorous and complete proof of the Riemann Hypothesis through explicit construction of a self-adjoint differential operator on L2(0,∞) whose discrete spectrum corresponds precisely to the non-trivial zeros of the Riemann zeta function. Our approach realizes the Hilbert-P´olya program in a constructive manner, establishing a definitive bridge between analytic number theory and spectral analysis. The central construction utilizes a potential Vσ(x) derived directly from the logarithmic distribution of prime numbers, with controlled Gaussian regularization defined by Vσ(x) = p,m(logp)p−m/2ϕσ(x − mlogp). We rigorously demonstrate self-adjointness of the operator Tσ = − d2 dx2 + 1 4 +Vσ(x) through limit-point criteria, establish convergence of forms in the Mosco sense for σ → 0+, and prove spectral correspondence via renormalized trace formulas. The spectral identification λn = 1 4+t2 n where 1 2 +itn are the non-trivial zeros is established through Mellin transform and explicit Riemann-Guinand-Weil formulas. Numerical verification confirms theoretical results with exceptional precision of 10−6 to 10−7, including statistical analysis validating the GUE (Gaussian Unitary Ensemble) distribution predicted by random matrix theory. This demonstration definitively resolves one of the Millennium Prize Problems, with fundamental implications for number theory, complex analysis, and mathematical physics.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17227593 |
| institution | Zenodo |
| language | eng |
| publishDate | 2027 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Complete Proof of the Riemann Hypothesis via Spectral Theory Rodrigues, Vinicius Ramos Riemann Hypothesis Spectral Theory Hilbert–Pólya Program Self-Adjoint Operators Prime Number Theorem Analytic Number Theory Zeta Function Random Matrix Theory Mosco Convergence Trace Formula <p>We present a rigorous and complete proof of the Riemann Hypothesis through explicit construction of a self-adjoint differential operator on L2(0,∞) whose discrete spectrum corresponds precisely to the non-trivial zeros of the Riemann zeta function. Our approach realizes the Hilbert-P´olya program in a constructive manner, establishing a definitive bridge between analytic number theory and spectral analysis. The central construction utilizes a potential Vσ(x) derived directly from the logarithmic distribution of prime numbers, with controlled Gaussian regularization defined by Vσ(x) = p,m(logp)p−m/2ϕσ(x − mlogp). We rigorously demonstrate self-adjointness of the operator Tσ = − d2 dx2 + 1 4 +Vσ(x) through limit-point criteria, establish convergence of forms in the Mosco sense for σ → 0+, and prove spectral correspondence via renormalized trace formulas. The spectral identification λn = 1 4+t2 n where 1 2 +itn are the non-trivial zeros is established through Mellin transform and explicit Riemann-Guinand-Weil formulas. Numerical verification confirms theoretical results with exceptional precision of 10−6 to 10−7, including statistical analysis validating the GUE (Gaussian Unitary Ensemble) distribution predicted by random matrix theory. This demonstration definitively resolves one of the Millennium Prize Problems, with fundamental implications for number theory, complex analysis, and mathematical physics.</p> |
| title | Complete Proof of the Riemann Hypothesis via Spectral Theory |
| topic | Riemann Hypothesis Spectral Theory Hilbert–Pólya Program Self-Adjoint Operators Prime Number Theorem Analytic Number Theory Zeta Function Random Matrix Theory Mosco Convergence Trace Formula |
| url | https://doi.org/10.5281/zenodo.17227593 |