Top: Predicted luminosity functions in the energy range 2–10 keV at low redshifts ﹩0.5< z< 1﹩ (dashed line) and high redshifts ﹩2< z< 4﹩ (solid line) compared with observational values derived from the same sample used in Fiore et al. (2003) to derive the densities in Fig. 8. Bottom: Predicted luminosity functions in the energy range 2–10 keV compared with the observational values obtained by Ueda et al. (2003) at various redshifts.