In the deepest optically identified X-ray survey yet performed, we have identified 32 X-ray selected QSOs to a flux limit of 2x10^{-15} erg cm^{-2} s^{-1} (0.5-2 keV). The survey, performed with the ROSAT PSPC, has 89% spectroscopic completeness. The QSO log(N)-log(S) relation is found to have a break to a flat slope at faint fluxes. The surface density of QSOs at the survey limit is 230+/-40 per square degree, the largest so far of any QSO survey. We have used this survey to measure the QSO X-ray luminosity function at low luminosities (Lx<10^{44.5} erg s^{-1}) and high redshifts (1<z<2.5). The highest redshift QSO in the survey has z=3.4. Combined with the QSOs from the Einstein EMSS at bright fluxes, we find pure luminosity evolution of the form Lx proportional to (1+z)^{3.0(+0.2,-0.3)} is an adequate description of the evolution of the X-ray luminosity function at low redshifts. A redshift cutoff in the evolution is required at z=1.4 ^{+0.4}_{-0.17} (for qo=0.5). We discuss the form of this evolution, its dependence on the model assumed and the errors on the derived parameters. We show that most previous X-ray surveys, including the EMSS, are consistent with a power law luminosity evolution index of 3.0. The contribution of QSOs to the 1-2 keV cosmic X-ray background is found to be between 31% and 51%.
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