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We consider the behaviour of a quantum scalar field on three-dimensionalEuclidean backgrounds: Anti-de Sitter space, the regular BTZ black holeinstanton and the BTZ instanton with a conical singularity at the horizon. Thecorresponding heat kernel and effective action are calculated explicitly forboth rotating and non-rotating holes. The quantum entropy of the BTZ black holeis calculated by differentiating the effective action with respect to theangular deficit at the conical singularity. The renormalization of theUV-divergent terms in the action and entropy is considered. The structure ofthe UV-finite term in the quantum entropy is of particular interest. Beingnegligible for large outer horizon area $A_+$ it behaves logarithmically forsmall $A_+$. Such behaviour might be important at late stages of black holeevaporation.Comment: 28 pages, latex, 2 figures now include

Quantum scalar field on a three-dimensional (BTZ) black hole instanton: Heat kernel, effective action, and thermodynamics