We present the hard-band ($2-10,mathrm{keV}$) X-ray luminosity function (HXLF) of $0.5-2,mathrm{keV}$ band selected AGN at high redshift. We have assembled a sample of 141 AGN at $3<zlesssim5$ from X-ray surveys of different size and depth, in order
to sample different regions in the $ L_X - z$ plane. The HXLF is fitted in the range $mathrm{logL_Xsim43-45}$ with standard analytical evolutionary models through a maximum likelihood procedure. The evolution of the HXLF is well described by a pure density evolution, with the AGN space density declining by a factor of $sim10$ from $z=3$ to 5. A luminosity-dependent density evolution model which, normally, best represents the HXLF evolution at lower redshift, is also consistent with the data, but a larger sample of low-luminosity ($mathrm{logL_X}<44$), high-redshift AGN is necessary to constrain this model. We also estimated the intrinsic fraction of AGN obscured by a column density $mathrm{logN_H}geq23$ to be $0.54pm0.05$, with no strong dependence on luminosity. This fraction is higher than the value in the Local Universe, suggesting an evolution of the luminous ($mathrm{L_X>10^{44}mathrm{erg,s^{-1}}}$) obscured AGN fraction from $z=0$ to $z>3$.
We have investigated the gas content of a sample of several hundred AGN host galaxies at z$<$1 and compared it with a sample of inactive galaxies, matched in bins of stellar mass and redshift. Gas masses have been inferred from the dust masses, obtai
ned by stacked Herschel far-IR and sub-mm data in the GOODS and COSMOS fields, under reasonable assumptions and metallicity scaling relations for the dust-to-gas ratio. We find that AGNs are on average hosted in galaxies much more gas rich than inactive galaxies. In the vast majority of stellar mass bins, the average gas content of AGN hosts is higher than in inactive galaxies. The difference is up to a factor of ten higher in low stellar mass galaxies, with a significance of 6.5$sigma$. In almost half of the AGN sample the gas content is three times higher than in the control sample of inactive galaxies. Our result strongly suggests that the probability of having an AGN activated is simply driven by the amount of gas in the host galaxy; this can be explained in simple terms of statistical probability of having a gas cloud falling into the gravitational potential of the black hole. The increased probability of an AGN being hosted by a star-forming galaxy, identified by previous works, may be a consequence of the relationship between gas content and AGN activity, found in this paper, combined with the Schmidt-Kennicutt law for star formation.
