  We express the zeta function associated to the Laplacian operator on
$S^1_r\times M$ in terms of the zeta function associated to the Laplacian on
$M$, where $M$ is a compact connected Riemannian manifold. This gives formulas
for the partition function of the associated physical model at low and high
temperature for any compact domain $M$. Furthermore, we provide an exact
formula for the zeta function at any value of $r$ when $M$ is a $D$-dimensional
box or a $D$-dimensional torus; this allows a rigorous calculation of the zeta
invariants and the analysis of the main thermodynamic functions associated to
the physical models at finite temperature.
