Statistical properties of photon count maps have recently been proven as a new tool to study the composition of the gamma-ray sky with high precision. We employ the 1-point probability distribution function of 6 years of Fermi-LAT data to measure the source-count distribution dN/dS and the diffuse components of the high-latitude gamma-ray sky as a function of energy. To that aim, we analyze the gamma-ray emission in five adjacent energy bands between 1 GeV and 171 GeV. It is demonstrated that the source-count distribution as a function of flux is compatible with a broken power law up to energies of ~50 GeV. The index below the break is between 1.95 and 2.0. For higher energies, a simple power-law fits the data, with an index of 2.2^{+0.7}_{-0.3} in the energy band between 50 GeV and 171 GeV. Upper limits on further possible breaks as well as the angular power of unresolved sources are derived. We find that point-source populations probed by this method can explain 83^{+7}_{-13}% (81^{+52}_{-19}%) of the extragalactic gamma-ray background between 1.04 GeV and 1.99 GeV (50 GeV and 171 GeV). The method has excellent capabilities for constraining the gamma-ray luminosity function and the spectra of unresolved blazars.
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