We present a determination of the distributions of gamma-ray photon flux -- the so called LogN-LogS relation -- and photon spectral index for blazars, based on the third extragalactic source catalog of the Fermi Gamma-ray Space Telescopes Large Area Telescope, and considering the photon energy range from 100 MeV to 100 GeV. The dataset consists of the 774 blazars in the so-called Clean sample detected with a greater than approximately seven sigma detection threshold and located above $pm$20 deg Galactic latitude. We use non-parametric methods verified in previous works to reconstruct the intrinsic distributions from the observed ones which account for the data truncations introduced by observational bias and includes the effects of the possible correlation between the flux and photon index. The intrinsic flux distribution can be represented by a broken power law with a high flux power-law index of -2.43$pm$0.08 and a low flux power-law index of -1.87$pm$0.10. The intrinsic photon index distribution can be represented by a Gaussian with mean of 2.62$pm$0.05 and width of 0.17$pm$0.02. We also report the intrinsic distributions for the sub-populations of BL Lac and FSRQ type blazars separately and these differ substantially. We then estimate the contribution of FSRQs and BL Lacs to the diffuse extragalactic gamma-ray background radiation. Under the simplistic assumption that the flux distributions probed in this analysis continue to arbitrary low flux, we calculate that the best fit contribution of FSRQs is 35% and BL Lacs 17% of the total gamma-ray output of the Universe in this energy range.
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