  The vacuum energy density (Casimir energy) corresponding to a massless scalar
quantum field living in different universes (mainly no-boundary ones), in
several dimensions, is calculated. Hawking's zeta function regularization
procedure supplemented with a very simple binomial expansion is shown to be a
rigorous and well suited method for performing the analysis. It is compared
with other, much more involved techniques. The principal-part prescription is
used to deal with the poles that eventually appear. Results of the analysis are
the absence of poles at four dimensions (for a 4d Riemann sphere and for a 4d
cylinder of 3d Riemann spherical section), the total coincidence of the results
corresponding to a 3d and a 4d cylinder (the first after pole subtraction), and
the fact that the vacuum energy density for cylinders is (in absolute value)
over an order of magnitude smaller than for spheres of the same dimension.
