ζ-function regularization of one-loop effective potentials in anti-de Sitter spacetime

We use the $\zeta$-function technique to calculate the one-loop effective potential for a scalar field in anti-de Sitter (AdS) space. The $\zeta$ function is computed exactly on the four-dimensional hyperbolic space $H^4$, the Euclidean section appropriate for AdS space. The structure of the ultraviolet divergences is shown to agree with previous calculations where Pauli-Villars or some version of dimensional regularization was used. The finite part of the effective potential is given explicitly by an integral over a variable related to the spectrum of the Laplace-Beltrami operator on $H^4$.