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The high-energy atmospheric neutrino flux is dominated by neutrinos from thedecays of charmed hadrons produced in the forward direction by cosmic rayinteractions with air nuclei. We evaluate the charm contributions to the promptatmospheric neutrino flux as a function of the center-of-mass energy $\sqrt{s}$of the hadronic collision and of the center-of-mass rapidity $y$ of theproduced charm hadron. Uncertainties associated with parton distributionfunctions are also evaluated as a function of $y$. We find that the $y$coverage of LHCb for forward heavy-flavour production, complemented by theangular coverage of present and future forward neutrino experiments at the LHC,bracket the most interesting $y$ regions for the prompt atmospheric neutrinoflux. At $\sqrt{s}=14$ TeV foreseen for the HL-LHC phase, nucleon collisions inair contribute to the prompt neutrino flux prominently below $E_\nu\sim10^7$~GeV. Measurements of forward charm and/or forward neutrinos produced inhadron collisions up to $\sqrt{s}=100$ TeV, which might become possible at theFCC, are relevant for the prompt atmospheric neutrino flux up to $E_\nu=10^8$GeV and beyond.

Forward production of prompt neutrinos from charm in the atmosphere and at high energy colliders