Holomorphic Hamiltonian $\xi$-Flow and Riemann Zeros

TL;DR

This paper explores a complex Hamiltonian framework inspired by quantum mechanics and trace formulas to analyze Riemann zeros, linking them to classical periodic orbits and spectral properties.

Contribution

It introduces a holomorphic Hamiltonian model connecting Riemann zeros with complex Hamiltonian flows and their quantization, offering a novel perspective on the zeros' spectral nature.

Findings

Hamiltonian phase portrait forms a Riemann surface related to Riemann zeros.

Flow map differential encodes all Riemann zeros.

Quantization yields a discrete spectrum tied to zeros' derivatives.

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 $H(q,p)=\xi(q)p$ 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.