X-ray Luminosity Function of Quasars at 3<z<5 from XMM-Newton
  Serendipitous Survey Data
  The X-ray luminosity function of distant (3<z<5.1) unabsorbed quasars has
been measured. A sample of distant high-luminosity quasars ($10^{45} \leq
L_{{\rm X},2-10} < 7.5 \times 10^{45}$ erg/s in the 2--10 keV energy band) from
the catalog given in Khorunzhev et al. (2016) compiled from the data of the
3XMM-DR4 catalog of the XMM-Newton serendipitous survey and the Sloan Digital
Sky Survey (SDSS) has been used. This sample consists of 101 sources. Most of
them (90) have spectroscopic redshifts $z_{spec}\geqslant 3$. The remaining
ones are quasar candidates with photometric redshift estimates
$z_{phot}\geqslant 3$. The spectroscopic redshifts of eight sources have been
measured with AZT-33IK and BTA telescopes. Owing to the record sky coverage
area ($\simeq 250$ sq. deg at X-ray fluxes $\sim 10^{-14}$ erg/s/cm$^{2}$ in
the 0.5-2 keV), from which the sample was drawn, we have managed to obtain
reliable estimates of the space density of distant X-ray quasars with
luminosities $L_{{\rm X},2-10} > 2 \times 10^{45}$ erg/s for the first time.
Their comoving space density remains constant as the redshift increases from
z=3 to z=5 to within a factor of 2. The power-law slope of the X-ray luminosity
function of high-redshift quasars in its bright end (above the break
luminosity) has been reliably constrained for the first time. The range of
possible slopes for the quasar luminosity and density evolution model is
$\gamma_2=2.78^{+0.00}_{-0.04}\pm0.20$, where initially the lower and upper
boundaries of $\gamma_2$ with the remaining uncertainty in the detection
completeness of X-ray sources in SDSS, and subsequently the statistical error
of the slope are specified.
