  Hattori-Itakura have recently derived the full Landau-level summation form
for the photon vacuum polarization tensor in constant external magnetic fields
at the one-loop level. The Landau-level summation form is essential when the
photon momentum exceeds the threshold of the pair creation of charged particles
in a magnetic field stronger than the squared mass of the charged particle. The
tensor has three different form factors depending on the tensor direction with
respect to the external magnetic field. The renormalization is nontrivial
because these form factors are expressed in terms of double or triple summation
forms. We give a numerical UV subtraction method which can be applied to
numerically evaluate the form factors in constant external magnetic fields. We
numerically investigate the photon vacuum polarization tensor in the form of
the Landau-level summation and estimate the systematic errors coming from
truncation of the Landau-level summation in a parameter region realized in
heavy ion collision experiments. We find that the error is practically
controllable at an $O(10^{-2})$ level for electrons and muons in strong
magnetic fields expected in heavy ion collisions in the experimentally feasible
kinematic parameter regions.
