  We consider the behaviour of a quantum scalar field on three-dimensional
Euclidean backgrounds: Anti-de Sitter space, the regular BTZ black hole
instanton and the BTZ instanton with a conical singularity at the horizon. The
corresponding heat kernel and effective action are calculated explicitly for
both rotating and non-rotating holes. The quantum entropy of the BTZ black hole
is calculated by differentiating the effective action with respect to the
angular deficit at the conical singularity. The renormalization of the
UV-divergent terms in the action and entropy is considered. The structure of
the UV-finite term in the quantum entropy is of particular interest. Being
negligible for large outer horizon area $A_+$ it behaves logarithmically for
small $A_+$. Such behaviour might be important at late stages of black hole
evaporation.
