  The two dimensional Coulomb gas is the prototypical model of statistical
mechanics displaying a special kind of phase transition, named after
Berezinskii, Kosterlitz and Thouless. Physicists and mathematicians proposed
several predictions about this system. Two of them, valid along the phase
transition curve and for small activity, are: a) the long-distance decay of the
"fractional charge" correlation is power law, with a multiplicative logarithmic
correction; b) in such a decay, the exponent of the power law, as well as the
exponent of the logarithmic correction, have a certain precise dependence upon
the charge value. In this paper we provide a proof of these two long standing
conjectures.
