  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
$\xi'(\rho_n)$ at Riemann zeros.
