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Neutrinos in the cosmic ray flux with energies near 1 EeV and above are detectable with the Surface Detector array of the Pierre Auger Observatory. We report here on searches through Auger data from 1 January 2004 until 20 June 2013. No neutrino cand idates were found, yielding a limit to the diffuse flux of ultra-high energy neutrinos that challenges the Waxman-Bahcall bound predictions. Neutrino identification is attempted using the broad time-structure of the signals expected in the SD stations, and is efficiently done for neutrinos of all flavors interacting in the atmosphere at large zenith angles, as well as for Earth-skimming neutrino interactions in the case of tau neutrinos. In this paper the searches for downward-going neutrinos in the zenith angle bins $60^circ-75^circ$ and $75^circ-90^circ$ as well as for upward-going neutrinos, are combined to give a single limit. The $90%$ C.L. single-flavor limit to the diffuse flux of ultra-high energy neutrinos with an $E^{-2}$ spectrum in the energy range $1.0 times 10^{17}$ eV - $2.5 times 10^{19}$ eV is $E_ u^2 dN_ u/dE_ u < 6.4 times 10^{-9}~ {rm GeV~ cm^{-2}~ s^{-1}~ sr^{-1}}$.
A measurement of the cosmic-ray spectrum for energies exceeding $4{times}10^{18}$ eV is presented, which is based on the analysis of showers with zenith angles greater than $60^{circ}$ detected with the Pierre Auger Observatory between 1 January 2004 and 31 December 2013. The measured spectrum confirms a flux suppression at the highest energies. Above $5.3{times}10^{18}$ eV, the ankle, the flux can be described by a power law $E^{-gamma}$ with index $gamma=2.70 pm 0.02 ,text{(stat)} pm 0.1,text{(sys)}$ followed by a smooth suppression region. For the energy ($E_text{s}$) at which the spectral flux has fallen to one-half of its extrapolated value in the absence of suppression, we find $E_text{s}=(5.12pm0.25,text{(stat)}^{+1.0}_{-1.2},text{(sys)}){times}10^{19}$ eV.
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