A Majorana Relativistic Quantum Spectral Approach to the Riemann Hypothesis in (1+1)-Dimensional Rindler Spacetimes

Abstract

Following the Hilbert-Pólya approach to the Riemann Hypothesis, we present an exact spectral realization of the nontrivial zeros of the Riemann zeta function $\zeta (z)$ 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 $\zeta (z)$ 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 $\zeta (z)$ 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.

73 pages, 2 columns, 84 references. Corrected typos in the text and in formulae, added in the appendix E a toy example of replacing ζ(2s) by the Dirichlet beta function $\beta (2s)$ validating the results with $\zeta (2s)$ function. Part of the logical procedure was tested also with Isabelle 2025