Zeta function continuation and the Casimir energy on odd and even dimensional spheres

The zeta function continuation method is applied to compute the Casimir energy on spheresS^N. Both odd and even dimensional spheres are studied. For the appropriate conformally modified Laplacian the Casimir energy is shown to be finite for all dimensions even though, generically, it should diverge in odd dimensions due to the presence of a pole in the associated zeta function (s). The residue of this pole is computed and shown to vanish in our case. An explicit analytic continuation of (s) is constructed for all values ofN. This enables us to calculate exactly and we find that the Casimir energy vanishes in all even dimensions. For odd dimensions is never zero but alternates in sign asN increases through odd values. Some results are also derived for the Casimir energy of other operators of Laplacian type.