Holomorphic Hamiltonian $\xi$-Flow and Riemann Zeros

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 $H(q,p)=\xi (q)p$ from the holomorphic flow $\dot{q}=\xi (q)$ and its variational differential equation. The Hamiltonian phase portrait $q(p)$ is a Riemann surface equivalent to reparameterized $\xi$-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 ${\xi }^{\mathrm{\prime }}({\rho }_n)$ at Riemann zeros.