2024/03/31 by Zeraoulia Rafik, Rafik, Zeraoulia, Caceres, Pedro
Mathematics · Physics and Astronomy · #47Sxx-19F05-37P99-39A22-15A33 #Advanced Mathematical Theories and Applications #Algebraic and Geometric Analysis #Chaotic Dynamics (nlin.CD) #FOS: Mathematics #FOS: Physical sciences #General Mathematics (math.GM) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2404.00583
openalex publication_date 2024/03/31 · openalex created_date 2024/04/04 · openalex updated_date 2026/07/28
This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis. Starting from the explicit zero-counting function N(T), we construct a discrete map on the critical line and analyse its Lyapunov exponents and bifurcation diagrams, showing that the smooth von Mangoldt term generates a strongly unstable flow that captures the global growth of the zero density. Motivated by this dynamics, we define a self-adjoint ``chaotic'' operator Oα on a weighted Hilbert space with weight dN/dT, prove its unboundedness and essential self-adjointness, and describe its spectral resolution via the spectral theorem. Finite-dimensional truncations of Oα yield Hermitian random matrices whose eigenvalue statistics agree numerically with Gaussian unitary ensemble predictions and show qualitative similarities to both Odlyzko's zeta zeros and the hydrogen-atom spectrum, suggesting that Oα lies in the same universality class as the nontrivial zeros and providing a concrete Hilbert--Pólya--type framework rather than a proof of the conjecture.