  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%.
