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

Application of abelian holonomy formalism to the elementary theory of numbers