Effective Finite Temperature Partition Function for Fields on Non-Commutative Flat Manifolds

The first quantum correction to the finite temperature partition function for a self-interacting massless scalar field on a $D-$dimensional flat manifold with $p$ non-commutative extra dimensions is evaluated by means of dimensional regularization, suplemented with zeta-function techniques. It is found that the zeta function associated with the effective one-loop operator may be nonregular at the origin. The important issue of the determination of the regularized vacuum energy, namely the first quantum correction to the energy in such case is discussed.