  A method to regularize and renormalize the fluctuations of a quantum field in
a curved background in the $\zeta$-function approach is presented. The method
produces finite quantities directly and finite scale-parametrized counterterms
at most. These finite couterterms are related to the presence of a particular
pole of the effective-action $\zeta$ function as well as to the heat kernel
coefficients. The method is checked in several examples obtaining known or
reasonable results. Finally, comments are given for as it concerns the recent
proposal by Frolov et al. to get the finite Bekenstein-Hawking entropy from
Sakharov's induced gravity theory.
