  A 'holographic formula' expressing the functional determinant of the
scattering operator in an asymptotically locally anti-de Sitter(ALAdS) space
has been proposed in terms of a relative functional determinant of the scalar
Laplacian in the bulk. It stems from considerations in AdS/CFT correspondence
of a quantum correction to the partition function in the bulk and the
corresponding subleading correction at large N on the boundary. In this paper
we probe this prediction for a class of quotients of hyperbolic space by a
discrete subgroup of isometries. We restrict to the simplest situation of an
abelian group where the quotient geometry describes thermal AdS and also the
non-spinning BTZ instanton. The bulk computation is explicitly done using the
method of images and the answer can be encoded in a (Patterson-)Selberg
zeta-function.
