Rigorous extension of the proof of zeta-function regularization

The proof of $\ensuremath{\zeta}$-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 $\ensuremath{\Sigma}{{n}_{1},\dots{},{n}_{N}=1}^{\ensuremath{\infty}}{(\ensuremath{\Sigma}{j=1}^{N}{a}_{j}{n}_{j}^{\ensuremath{\alpha}})}^{\ensuremath{-}s}$, $\ensuremath{\alpha}=2, 4, \dots{}$. 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 $\ensuremath{\alpha}$. The extra terms are calculated explicitly for any value of $\ensuremath{\alpha}\ensuremath{\le}2$. As an application, we provide the finite results corresponding to the $\ensuremath{\zeta}$-function regularization of expressions associated with field theories evaluated in partially compactified, toroidal spacetimes of the form ${\mathrm{T}}^{p}\ifmmode\times\else\texttimes\fi{}{\mathrm{R}}^{q+1}$