  We emphasize the close relationship between zeta function methods and
arbitrary spectral cutoff regularizations in curved spacetime. This yields, on
the one hand, a physically sound and mathematically rigorous justification of
the standard zeta function regularization at one loop and, on the other hand, a
natural generalization of this method to higher loops. In particular, to any
Feynman diagram is associated a generalized meromorphic zeta function. For the
one-loop vacuum diagram, it is directly related to the usual spectral zeta
function. To any loop order, the renormalized amplitudes can be read off from
the pole structure of the generalized zeta functions. We focus on scalar field
theories and illustrate the general formalism by explicit calculations at
one-loop and two-loop orders, including a two-loop evaluation of the conformal
anomaly.
