  We consider an abelian holonomy operator in two-dimensional conformal field
theory with zero-mode contributions. The analysis is made possible by use of a
geometric-quantization scheme for abelian Chern-Simons theory on $S^1 \times
S^1 \times {\bf R}$. We find that a purely zero-mode part of the holonomy
operator can be expressed in terms of Riemann's zeta function. We also show
that a generalization of linking numbers can be obtained in terms of the vacuum
expectation values of the zero-mode holonomy operators. Inspired by
mathematical analogies between linking numbers and Legendre symbols, we then
apply these results to a space of ${\bf F}_p = {\bf Z}/ p {\bf Z}$ where $p$ is
an odd prime number. This enables us to calculate "scattering amplitudes" of
identical odd primes in the holonomy formalism. In this framework, the Riemann
hypothesis can be interpreted by means of a physically obvious fact, i.e.,
there is no notion of "scattering" for a single-particle system. Abelian gauge
theories described by the zero-mode holonomy operators will be useful for
studies on quantum aspects of topology and number theory.
