  We apply the Sudakov technique to description of the multiple production of
lepton pairs in peripheral collisions of ultrarelativistic heavy ions. For
heavy ions with $Z_1\alpha\sim Z_2\alpha\ll 1$ one needs a careful treatment of
the multiple Coulomb exchange between colliding ions and screening effects,
whereas interaction of real or virtual lepton pairs with colliding ions can be
neglected. We demonstrate that while the inclusive spectra are modified by
multiple Coulomb exchange between the colliding ions, the Coulomb corrections
to the momentum integrated multiplicity distributions do vanish. After
transformation to the impact parameter representation the probability of $n$
lepton pair production is shown to obey the Poisson distribution. The relevant
cross section is obtained.
