  The purpose of this paper is to present a quantum statistical theory of
2-dimensional vortex gas based on the generalized Hamiltonian dynamics recently
developed. The quantized spectrum is evaluated for a pair of vortex on the
basis of the semiclassical quantization rule. This is used to evaluate the
partition function for a dilute vortex gas. A remarkable consequence is that
the partition function and related quantities are given in terms of the
generalized Riemann zeta function. The topological phase transition is
naturally understood as the pole structure of the zeta function.
