  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.
