2001/11/26 by Matti Pitkänen, Matti Pitkanen, Pitkanen, Matti
Chemistry · Mathematics · Physics and Astronomy · #FOS: Mathematics #General Mathematics (math.GM) #Molecular spectroscopy and chirality #Quantum Mechanics and Applications #Quantum chaos and dynamical systems #math.GM
paper · pdf · doi:10.48550/arxiv.math/0111262
14 pages, realization that positive definiteness of the metric is possible and also implies Riemann hypothesis, Lie group theoretic analytic argument proving Riemann hypothesis
openalex publication_date 2001/11/26 · arxiv created 2002/01/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A strategy for proving Riemann hypothesis is suggested. The vanishing of the Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D+ having the zeros of Riemann Zeta as its eigenvalues. The construction of D+ is inspired by the conviction that Riemann Zeta is associated with a physical system allowing conformal transformations as its symmetries. The eigenfunctions of D+ are analogous to the so called coherent states and in general not orthogonal to each other. The states orthogonal to a vacuum state (which has a negative norm squared) correspond to the zeros of the Riemann Zeta. The induced metric in the space \calV of states which correspond to the zeros of the Riemann Zeta at the critical line Re[s]=1/2 is hermitian and both hermiticity and positive definiteness properties imply Riemann hypothesis. Conformal invariance in the sense of gauge invariance allows only the states belonging to \calV. Riemann hypothesis follows also from a restricted form of a dynamical conformal invariance in \calV and one can reduce the proof to a standard analytic argument used in Lie group theory.