  We consider a two-dimensional Coulomb gas of positive and negative pointlike
unit charges interacting via a logarithmic potential. The density (rather than
the charge) correlation functions are studied. In the bulk, the form-factor
theory of an equivalent sine-Gordon model is used to determine the density
correlation length. At the surface of a rectilinear plain wall, the
universality of the asymptotic behavior of the density correlations is
suggested. A scaling analysis implies a local form of the compressibility sum
rule near a hard wall. A symmetry of the Coulomb system with respect to the
M\"obius conformal transformation, which induces a gravitational source acting
on the particle density, is established. Among the consequences, a universal
term of the finite-size expansion of the grand potential is derived exactly for
a disk geometry of the confining domain.
