  We present a measurement of the density of GeV muons in near-vertical air
showers using three years of data recorded by the IceTop array at the South
Pole. Depending on the shower size, the muon densities have been measured at
lateral distances between 200 m and 1000 m. From these lateral distributions,
we derive the muon densities as functions of energy at reference distances of
600 m and 800 m for primary energies between 2.5 PeV and 40 PeV and between 9
PeV and 120 PeV, respectively. The muon densities are determined using, as a
baseline, the hadronic interaction model Sibyll 2.1 together with various
composition models. The measurements are consistent with the predicted muon
densities within these baseline interaction and composition models. The
measured muon densities have also been compared to simulations using the
post-LHC models EPOS-LHC and QGSJet-II.04. The result of this comparison is
that the post-LHC models together with any given composition model yield higher
muon densities than observed. This is in contrast to the observations above 1
EeV where all model simulations yield for any mass composition lower muon
densities than the measured ones. The post-LHC models in general feature higher
muon densities so that the agreement with experimental data at the highest
energies is improved but the muon densities are not correct in the energy range
between 2.5 PeV and about 100 PeV.
