Equation of state in the fugacity format for the two-dimensional Coulomb gas

We derive the exact general form of the equation of state, in the fugacity format, for the two-dimensional Coulomb gas. Our results are valid in the conducting phase of the Coulomb gas, for temperatures above the Kosterlitz-Thouless transition. The derivation of the equation of state is based on the knowledge of the general form of the short-distance expansion of the correlation functions of the Coulomb gas. We explicitly compute the expansion up to order $O(\zeta^6)$ in the activity $\zeta$. Our results are in very good agreement with Monte Carlo simulations at very low density.