We consider various regularization schemes for calculating the renormalized vacuum energy of a massless scalar field in the n-dimensional Einstein universe. We also study a related problem, namely, the Casimir energy for a massless scalar field in the n-dimensional Einstein universe subject to Dirichlet boundary conditions on a sphere of maximal radius. In a recent work the author used the representation theory of SO(2,n) to obtain exact results but not in closed form for the second problem with n arbitrary. Here we make use of generating functions for generalized Bernoulli polynomials and an extension of a result of Srivastava and Todorov about generalized Bernoulli numbers (Srivastava, Todorov, J. Math. Anal. Appl. 130:509-513, 1988) to obtain new results involving exact expressions in closed form for both problems. We also consider expansions of the generalized Bernoulli polynomials into Hurwitz zeta functions which enables us to explicitly demonstrate the equivalence of the cutoff function technique with the zeta regularization technique. Our method of approach confirms the results of Herdeiro et al. (Class. Quant. Gravit. 25:165010, 2008) and Özcan (Class. Quant. Gravit. 23:5531-5546, 2006).We conclude the paper by showing that useful information about the analogous problem in n-dimensional Minkowski space can also be obtained out of our analysis.