[en] We improve the accuracy of the extrapolation of the gluon distribution of the proton to very small x, and show that the charm production cross section, needed to calculate the cosmic ray-induced 'atmospheric' flux of ultrahigh energy prompt ν_μ and ν_τ neutrinos, may be predicted within perturbative QCD to within about a factor of three. We follow the sequence of interactions and decays in order to calculate the neutrino fluxes as a function of energy up to 10⁹ GeV. We also compute the prompt ν_τ flux from bb-bar production, hadronization and decay. New cosmic sources of neutrinos will be indicated if more prompt neutrinos are observed than predicted. If fewer neutrinos are observed than predicted, then constraints will be imposed on the nuclear composition of cosmic rays. The advantages of studying ν_τ neutrinos are emphasized. We provide a simple parametrization of the prediction for the inclusive cross section for c quark production in high energy proton-air collisions. (author)