Prompt neutrinos from atmospheric c-cbar and b-bbar production and the gluon at very small x

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 muon and tau 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 10^9 GeV. We also compute the prompt neutrino tau flux from b-bbar 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 tau neutrinos are emphasized. We provide a simple parameterization of the prediction for the inclusive cross section for c quark production in high energy proton--air collisions.