  The off-shell entropy for a massless scalar field in a D-dimensional
Rindler-like space-time is investigated within the conical Euclidean approach
in the manifold $C_\be\times\M^N$, $C_\be$ being the 2-dimensional cone, making
use of the zeta-function regularisation. Due to the presence of conical
singularities, it is shown that the relation between the zeta-function and the
heat kernel is non trivial and, as first pointed out by Cheeger, requires a
separation between small and large eigenvalues of the Laplace operator. As a
consequence, in the massless case, the (naive) non existence of the Mellin
transform is by-passed by the Cheeger's analytical continuation of the
zeta-function on manifold with conical singularities. Furthermore, the
continuous spectrum leads to the introduction of smeared traces. In general, it
is pointed out that the presence of the divergences may depend on the smearing
function and they arise in removing the smearing cutoff. With a simple choice
of the smearing function, horizon divergences in the thermodynamical quantities
are recovered and these are similar to the divergences found by means of
off-shell methods like the brick wall model, the optical conformal
transformation techniques or the canonical path integral method.
