  We review the application of the spectral zeta-function to the 1- loop
properties of quantum field theories on manifolds with boundary, with emphasis
on Euclidean quantum gravity and quantum cosmology. As was shown in the
literature some time ago, the only boundary conditions that are completely
invariant under infinitesimal diffeomorphisms on metric perturbations suffer
from a drawback, i.e. lack of strong ellipticity of the resulting
boundary-value problem. Nevertheless, at least on the Euclidean 4-ball
background, it remains possible to evaluate the zeta(0) value, which describes
in this case a universe which, in the limit of small 3-geometry, has vanishing
probability of approaching the cosmological singularity. An assessment of this
result is here performed, discussing its physical and mathematical
implications.
