ζ-function regularization of the effective action for a δ-function potential

We present an exact evaluation of the generalized ζ-function for a real scalar field in a δ-function potential in the spacetime S1×RD−1. The result for the ζ-function is used to obtain the effective action. As a byproduct of the calculation the heat-kernel coefficients are obtained to all orders in a closed form. The regularized zero-point energy is found and we discuss the renormalization of the effective action.