  We present the results of the analysis of the AGN population in the deepest
extragalactic hard X-ray survey. The survey is based on INTEGRAL observation of
the 3C 273/Coma cluster region, and covers 2500 deg2 with a 20-60 keV flux
limit 1.5 times lower than other surveys at similar energies, resolving about
2.5% of the cosmic hard X-ray background. Using this survey, we can constrain
in an unbiased way the distribution of hydrogen column absorption up to
Nh=10^25 cm-2. We put an upper limit of 24% to the fraction of Compton-thick
objects. Compared to models of the AGN population selected in the 2-10 keV
band, the Log N-Log S diagram is generally in good agreement, but the Nh
distribution is significantly different, with significantly less unabsorbed
sources (Nh<10^22 cm-2) at a given flux limit compared to the models. We also
study the local hard X-ray luminosity function (LF), which is compatible with
what is found in other recent hard X-ray surveys. The extrapolation of the 2-10
keV LF is lower than the hard X-ray LF. The discrepancy is resolved if AGN
spectra typically present reflection humps with reflection fraction R~1.
Finally, we use the population properties of this survey to show that a future
ultra-deep INTEGRAL extragalactic survey can result in a quite large AGN sample
with enough objects at redshifts larger than z=0.05 so that we can detect
evolution in the hard X-ray LF.
