  We prove convergence of the multi-body correlation function as a power series
in the density. We work in the context of the cluster expansion in the
canonical ensemble and we obtain bounds uniform in the volume and the number of
particles. In the thermodynamic limit, the coefficients are characterized by
sums over some class of two-connected graphs. We introduce the "direct
correlation function" in the canonical ensemble and we prove that in the
thermodynamic limit it is given by a convergent power series in the density
with coefficients given by sums over some other class of two-connected graphs.
Furthermore, it satisfies the Ornstein-Zernike equation from which quantified
approximations can be derived.
