Zeta function regularization for a scalar field in a compact domain

We express the zeta function associated with the Laplacian operator on S¹_r × M in terms of the zeta function associated with the Laplacian on M, where M is a compact connected Riemannian manifold. This gives formulae 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 with the physical models at finite temperature.