  We investigate the possibility that alpha_s freezes as function of N_f within
perturbation theory. We use two approaches -- direct search for a zero in the
effective-charge (ECH) beta function, and the Banks-Zaks (BZ) expansion. We
emphasize the fundamental difference between quantities with space-like vs.
those with time-like momentum. We show that within the ECH approach several
space-like quantities exhibit similar behavior. In general the 3-loop ECH beta
functions can lead to freezing for N_f\gsim 5, but higher-order calculations
are essential for a conclusive answer. The BZ expansion behaves differently for
different observables. Assuming that the existence of a fixed point requires
convergence of the BZ expansion for any observable, we can be pretty sure that
there is no fixed point for N_f\lsim 12. The consequences of the Crewther
relation concerning perturbative freezing are analyzed. We also emphasize that
time-like quantities have a consistent infrared limit only when the
corresponding space-like effective charge has one. We show that perturbative
freezing can lead to an analyticity structure in the complex momentum-squared
plane that is consistent with causality.
