A Majorana Relativistic Quantum Spectral Approach to the Riemann Hypothesis in (1+1)-Dimensional Rindler Spacetimes
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916854926147584 |
|---|---|
| author | Tamburini, Fabrizio |
| author_facet | Tamburini, Fabrizio |
| contents | Following the Hilbert-Pólya approach to the Riemann Hypothesis, we present an exact spectral realization of the nontrivial zeros of the Riemann zeta function $ζ(z)$ with a Mellin-Barnes integral that explicitly contains it. This integral defines the spectrum of the real-valued energy eigenvalues $E_n$ of a Majorana particle in a $(1+1)$-dimensional Rindler spacetime or equivalent Kaluza-Klein reductions of $(n+1)$-dimensional geometries. We show that the Hamiltonian $H_M$ describing the particle is hermitian and the spectrum of energy eigenvalues $\{E_n\}_{n \in \mathbb{N}}$ is countably infinite in number in a bijective correspondence with the imaginary part of the nontrivial zeros of $ζ(z)$ having the same cardinality as required by Hardy-Littlewood's theorem from number theory. The correspondence between the two spectra with the essential self-adjointness of $H_M$, confirmed with deficiency index analysis, boundary triplet theory and Krein's extension theorem, imply that all nontrivial zeros have real part $\Re ( z )=1/2$, i.e., lie on the ``critical line''. In the framework of noncommutative geometry, $H_M$ is interpreted as a Dirac operator $D$ in a spectral triple $(\mathcal{A}, \mathcal{H}, D)$, linking these results to Connes' program for the Riemann Hypothesis. The algebra $\mathcal{A}$ encodes the modular symmetries underlying the spectral realization of $ζ(z)$ in the Hilbert space $\mathcal{H}$ of Majorana wavefunctions, integrating concepts from quantum mechanics, general relativity, and number theory. This analysis offers a promising Hilbert-Pólya-inspired path to prove the Riemann Hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Majorana Relativistic Quantum Spectral Approach to the Riemann Hypothesis in (1+1)-Dimensional Rindler Spacetimes Tamburini, Fabrizio General Mathematics High Energy Physics - Theory Following the Hilbert-Pólya approach to the Riemann Hypothesis, we present an exact spectral realization of the nontrivial zeros of the Riemann zeta function $ζ(z)$ with a Mellin-Barnes integral that explicitly contains it. This integral defines the spectrum of the real-valued energy eigenvalues $E_n$ of a Majorana particle in a $(1+1)$-dimensional Rindler spacetime or equivalent Kaluza-Klein reductions of $(n+1)$-dimensional geometries. We show that the Hamiltonian $H_M$ describing the particle is hermitian and the spectrum of energy eigenvalues $\{E_n\}_{n \in \mathbb{N}}$ is countably infinite in number in a bijective correspondence with the imaginary part of the nontrivial zeros of $ζ(z)$ having the same cardinality as required by Hardy-Littlewood's theorem from number theory. The correspondence between the two spectra with the essential self-adjointness of $H_M$, confirmed with deficiency index analysis, boundary triplet theory and Krein's extension theorem, imply that all nontrivial zeros have real part $\Re ( z )=1/2$, i.e., lie on the ``critical line''. In the framework of noncommutative geometry, $H_M$ is interpreted as a Dirac operator $D$ in a spectral triple $(\mathcal{A}, \mathcal{H}, D)$, linking these results to Connes' program for the Riemann Hypothesis. The algebra $\mathcal{A}$ encodes the modular symmetries underlying the spectral realization of $ζ(z)$ in the Hilbert space $\mathcal{H}$ of Majorana wavefunctions, integrating concepts from quantum mechanics, general relativity, and number theory. This analysis offers a promising Hilbert-Pólya-inspired path to prove the Riemann Hypothesis. |
| title | A Majorana Relativistic Quantum Spectral Approach to the Riemann Hypothesis in (1+1)-Dimensional Rindler Spacetimes |
| topic | General Mathematics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2503.09644 |