  We evaluate the prompt atmospheric neutrino flux at high energies using three
different frameworks for calculating the heavy quark production cross section
in QCD: NLO perturbative QCD, $k_T$ factorization including low-$x$
resummation, and the dipole model including parton saturation. We use QCD
parameters, the value for the charm quark mass and the range for the
factorization and renormalization scales that provide the best description of
the total charm cross section measured at fixed target experiments, at RHIC and
at LHC. Using these parameters we calculate differential cross sections for
charm and bottom production and compare with the latest data on forward charm
meson production from LHCb at $7$ TeV and at $13$ TeV, finding good agreement
with the data. In addition, we investigate the role of nuclear shadowing by
including nuclear parton distribution functions (PDF) for the target air
nucleus using two different nuclear PDF schemes. Depending on the scheme used,
we find the reduction of the flux due to nuclear effects varies from $10\%$ to
$50 \%$ at the highest energies. Finally, we compare our results with the
IceCube limit on the prompt neutrino flux, which is already providing valuable
information about some of the QCD models.
