Scalar and spinor Casimir energies in even-dimensional Kaluza-Klein spaces of the form ${\mathit{M}}^4$×${\mathit{S}}^{\mathit{N}1}$×${\mathit{S}}^{\mathit{N}2}$×⋅⋅⋅

We use two methods of computing the unique logarithmically divergent part of the Casimir energy for massive scalar and spinor fields defined on even-dimensional Kaluza-Klein spaces of the form ${\mathit{M}}^4$×${\mathit{S}}^{\mathit{N}1}$×${\mathit{S}}^{\mathit{N}2}$×⋅⋅⋅. Both methods (heat kernel and direct) give identical results. The first evaluates the required internal ζ function by identifying it in the asymptotic expansion of the trace of the heat kernel, and the second evaluates the ζ function directly using the Euler-Maclaurin sum formula. In Appendix C we tabulate these energies for all spaces of total internal dimension ≤6. These methods are easily applied to vector and tensor fields needed in computing one-loop vacuum gravitational energies on these spaces. Stable solutions are given for internal structure $S^2$×$S^2$.