Rigorous extension of the proof of zeta-function regularization

The proof of $\zeta$-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 $\Sigma {n_1,\dots ,n_N=1\mathrm{}}^{\infty }{\left(\Sigma {j=1\mathrm{}}^N{a}_j{n}_j^{\alpha }\right)}^{-s}$, $\alpha =2, 4, \dots$. 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 $\alpha$. The extra terms are calculated explicitly for any value of $\alpha \le 2$. As an application, we provide the finite results corresponding to the $\zeta$-function regularization of expressions associated with field theories evaluated in partially compactified, toroidal spacetimes of the form ${\mathrm{T}}^p\times {\mathrm{R}}^{q+1\mathrm{}}$.