  One-loop Standard Model observables produced by virtual heavy Kaluza-Klein
fields play a prominent role in the minimal model of universal extra
dimensions. Motivated by this aspect, we integrate out all the Kaluza-Klein
heavy modes coming from the Yang-Mills theory set on a spacetime with an
arbitrary number, $n$, of compact extra dimensions. After fixing the gauge with
respect to the Kaluza-Klein heavy gauge modes in a covariant manner, we
calculate a gauge independent effective Lagrangian expansion containing
multiple Kaluza-Klein sums that entail a bad divergent behavior. We use the
Epstein-zeta function to regularize and characterize discrete divergences
within such multiple sums, and then we discuss the interplay between the number
of extra dimensions and the degree of accuracy of effective Lagrangians to
generate or not divergent terms of discrete origin. We find that
nonrenormalizable terms with mass dimension $k$ are finite as long as $k>4+n$.
Multiple Kaluza-Klein sums of nondecoupling logarithmic terms, not treatable by
Epstein-zeta regularization, are produced by four-dimensional momentum
integration. On the grounds of standard renormalization, we argue that such
effects are unobservable.
