Around a rapidly rotating black hole (BH), when the plasma accretion rate is much less than the Eddington rate, the radiatively inefficient accretion flow (RIAF) cannot supply enough MeV photons that are capable of materializing as pairs. In such a charge-starved BH magnetosphere, the force-free condition breaks down in the polar funnels. Applying the pulsar outer-magnetospheric lepton accelerator theory to super-massive BHs, we demonstrate that a strong electric field arises along the magnetic field lines in the direct vicinity of the event horizon in the funnels, that the electrons and positrons are accelerated up to 100~TeV in this vacuum gap, and that these leptons emit copious photons via inverse-Compton (IC) process between 0.1~TeV and 30~TeV for a distant observer. It is found that these IC fluxes will be detectable with Imaging Atmospheric Cherenkov Telescopes, provided that a low-luminosity active galactic nucleus is located within 1~Mpc for a million-solar-mass central BH or within 30~Mpc for a billion-solar-mass central BH. These very-high-energy fluxes are beamed in a relatively small solid angle around the rotation axis because of the inhomogeneous and anisotropic distribution of the RIAF photon field, and show an anti-correlation with the RIAF submillimeter fluxes. The gap luminosity little depends on the three-dimensional magnetic-field configuration, because the Goldreich-Julian charge density, and hence the exerted electric field is essentially governed by the frame-dragging effect, not by the magnetic field configuration.
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