Rigorous extension of the proof of zeta-function regularization

The proof of ζ-function regularization of high-temperature expansions, a technique which provides correct results for many field-theoretical quantities of interest, is known to fail, however, in the case of ''Epstein-type'' expressions such as [Σ/sup ∞//sub n//sub 1,...,n//sub N/=1(Σ/sup N//sub j=1/a/sub j/j/sup α/)/sup -s/, α=2,4,... . After showing where precisely the existing demonstration breaks down, we provide a new proof of this regularization valid for a wider range of the parameter α. The extra terms are calculated explicitly for any value of α ≤ 2. As an application, we provide the finite results corresponding to the ζ-function regularization of expressions associated with field theories evaluated in partially compactified, toroidal spacetimes of the form T/sup p/ x R/sup q+1/