Holomorphic Hamiltonian ξ-Flow and Riemann Zeros

2020/06/16 by Dirk Lebiedz, Lebiedz, Dirk
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Meromorphic and Entire Functions #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2006.09165

openalex publication_date 2020/06/16 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

With a view on the formal analogy between Riemann-von-Mangoldts explicit formula and semiclassical quantum mechanics in terms of the Gutzwiller trace formula we construct a complex-valued Hamiltonian H(q,p)=ξ(q)p from the holomorphic flow q=ξ(q) and its variational differential equation. The Hamiltonian phase portrait q(p) is a Riemann surface equivalent to reparameterized ξ-Newton flow solutions in complex-time, its flow map differential is determined by all Riemann zeros and reminiscent of a 'spectral sum' in trace formulas. Canonical quantization for particle quantum mechanics on a circle leads to a Dirac-type momentum operator with discrete spectrum given by classical closed orbit periods determined by derivatives ξ'(ρ_n) at Riemann zeros.

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