  Making use of inverse Mellin transform techniques for analytical
continuation, an elegant proof and an extension of the zeta function
regularization theorem is obtained. No series commutations are involved in the
procedure; nevertheless the result is naturally split into the same three
contributions of very different nature, i.e. the series of Riemann zeta
functions and the power and negative exponentially behaved functions,
respectively, well known from the original proof. The new theorem deals equally
well with elliptic differential operators whose spectrum is not explicitly
known. Rigorous results on the asymptoticity of the outcoming series are given,
together with some specific examples. Exact analytical formulas, simplifying
approximations and numerical estimates for the last of the three contributions
(the most difficult to handle) are obtained. As an application of the method,
the summation of the series which appear in the analytic computation (for
different ranges of temperature) of the partition function of the string
---basic in order to ascertain if QCD is some limit of a string theory--- is
PERFORMED.
