Speculative zeta-function connection for quantized vortex dynamics

Let $(S,g)$ be a compact Riemann surface, let $G$ be the Green function and $R$ its associated Robin function, and let $s_o$ denote a vortex position. Consider the vortex monopole system

$$\dot{s} = {\rm sgrad}\,G(s,s_o),\qquad \dot{s}_o = {\rm sgrad}\,R(s_o),$$

or a vortex pair with Hamiltonian

$$H=-\kappa^2\frac{\log d(s_1,s_2)}{2\pi}+\kappa^2B(s_1,s_2).$$

Speculative zeta-function connection. A connection may exist between the zeros of Riemann's zeta function and the quantization of a $3/2$ degrees-of-freedom vortex monopole problem on a compact Riemann surface with metric $g$, either in the monopole system above or in the vortex-pair system. This proposal is explicitly speculative and is not presented as an established mathematical conjecture; the supplied text gives no evidence that such a connection has been proved or disproved.