The Fermi-LAT high-latitude Survey: Source Count Distributions and the
  Origin of the Extragalactic Diffuse Background
  This is the first of a series of papers aimed at characterizing the
populations detected in the high-latitude sky of the {\it Fermi}-LAT survey. In
this work we focus on the intrinsic spectral and flux properties of the source
sample. We show that when selection effects are properly taken into account,
{\it Fermi} sources are on average steeper than previously found (e.g. in the
bright source list) with an average photon index of 2.40$\pm0.02$ over the
entire 0.1--100\,GeV energy band. We confirm that FSRQs have steeper spectra
than BL Lac objects with an average index of 2.48$\pm0.02$ versus
2.18$\pm0.02$. Using several methods we build the deepest source count
distribution at GeV energies deriving that the intrinsic source (i.e. blazar)
surface density at F$_{100}\geq10^{-9}$\,ph cm$^{-2}$ s$^{-1}$ is
0.12$^{+0.03}_{-0.02}$\,deg$^{-2}$. The integration of the source count
distribution yields that point sources contribute 16$(\pm1.8)$\,\% ($\pm$7\,\%
systematic uncertainty) of the GeV isotropic diffuse background. At the fluxes
currently reached by LAT we can rule out the hypothesis that point-like sources
(i.e. blazars) produce a larger fraction of the diffuse emission.
