  We investigate the population of high-redshift ($3\leq z < 6$) AGN selected
in the two deepest X-ray surveys, the 7 Ms \textit{Chandra} Deep Field-South
and 2 Ms \textit{Chandra} Deep Field-North. Their outstanding sensitivity and
spectral characterization of faint sources allow us to focus on the sub-$L_*$
regime (log$L_{\mathrm{X}}\lesssim44$), poorly sampled by previous works using
shallower data, and the obscured population. Taking fully into account the
individual photometric-redshift probability distribution functions, the final
sample consists of $\approx102$ X-ray selected AGN at $3\leq z < 6$. The
fraction of AGN obscured by column densities log$N_{\mathrm{H}}>23$ is
$\sim0.6-0.8$, once incompleteness effects are taken into account, with no
strong dependence on redshift or luminosity. We derived the high-redshift AGN
number counts down to
$F_{\mathrm{0.5-2\,keV}}=7\times10^{-18}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$,
extending previous results to fainter fluxes, especially at $z>4$. We put the
tightest constraints to date on the low-luminosity end of AGN luminosity
function at high redshift. The space-density, in particular, declines at $z>3$
at all luminosities, with only a marginally steeper slope for low-luminosity
AGN. By comparing the evolution of the AGN and galaxy densities, we suggest
that such a decline at high luminosities is mainly driven by the underlying
galaxy population, while at low luminosities there are hints of an intrinsic
evolution of the parameters driving nuclear activity. Also, the black-hole
accretion rate density and star-formation rate density, which are usually found
to evolve similarly at $z\lesssim3$, appear to diverge at higher redshifts.
