Vacuum Polarization in Quantum Electrodynamics

It is shown that if the photon propagator is set equal to $\frac{1}{k^2}$ in the expression for the vacuum polarization, then $Z_3^{}{}_{}{}^{-1\mathrm{}}$ (where $Z_3$ is the photon wave-function renormalization constant) diverges like a single power of the logarithm of an ultraviolet cutoff in all orders of perturbation theory. The implication of this result upon the possible finiteness of ordinary quantum electrodynamics is discussed.