A systematic study of the properties of particle and charge correlation functions in the two-dimensional Coulomb gas confined to a one-dimensional domain is undertaken. Two versions of this system are considered: one in which the positive and negative charges are constrained to alternate in sign along the line, and the other where there is no charge ordering constraint. Both systems undergo a zero-density Kosterlitz-Thouless type transition as the dimensionless coupling $Γ:= q^2 / kT$ is varied through $Γ= 2$. In the charge ordered system we use a perturbation technique to establish an $O(1/r^4)$ decay of the two-body correlations in the high temperature limit. For $Γ\rightarrow 2^+$, the low-fugacity expansion of the asymptotic charge-charge correlation can be resummed to all orders in the fugacity. The resummation leads to the Kosterlitz renormalization equations.

39 pages, 5 figures not included, Latex, to appear J. Stat. Phys. Shortened version of abstract below