  Vacuum polarization of charged massless fermions is investigated in the
superposition of Coulomb and Aharonov--Bohm (AB) potentials in 2+1 dimensions.
For this purpose we construct the Green function of the two-dimensional Dirac
equation with Coulomb and AB potentials (via the regular and irregular
solutions of the radial Dirac equation) and calculate the vacuum polarization
charge density in these fields in the so-called subcritical and supercritical
regimes.
  The role of the self-adjoint extension parameter is discussed in terms of the
physics of problem. We hope that our results will be helpful in the more deep
understanding the fundamental problem of quantum electrodynamics and can be
applied to the problems of charged impurity screening in graphene with taking
into consideration the electron spin.
