2025/03/12 by Tamburini, Fabrizio · 1 citation
#FOS: Mathematics #FOS: Physical sciences #General Mathematics (math.GM) #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.2503.09644
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 En 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 HM describing the particle is hermitian and the spectrum of energy eigenvalues \En\n ∈ 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 HM, 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, HM is interpreted as a Dirac operator D in a spectral triple (A, H, D), linking these results to Connes' program for the Riemann Hypothesis. The algebra A encodes the modular symmetries underlying the spectral realization of ζ(z) in the Hilbert space 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.