The Final SDSS High-Redshift Quasar Sample of 52 Quasars at z>5.7
  We present the discovery of nine quasars at $z\sim6$ identified in the Sloan
Digital Sky Survey (SDSS) imaging data. This completes our survey of $z\sim6$
quasars in the SDSS footprint. Our final sample consists of 52 quasars at
$5.7<z\le6.4$, including 29 quasars with $z_{\rm AB}\le20$ mag selected from
11,240 deg$^2$ of the SDSS single-epoch imaging survey (the main survey), 10
quasars with $20\le z_{\rm AB}\le20.5$ selected from 4223 deg$^2$ of the SDSS
overlap regions (regions with two or more imaging scans), and 13 quasars down
to $z_{\rm AB}\approx22$ mag from the 277 deg$^2$ in Stripe 82. They span a
wide luminosity range of $-29.0\le M_{1450}\le-24.5$. This well-defined sample
is used to derive the quasar luminosity function (QLF) at $z\sim6$. After
combining our SDSS sample with two faint ($M_{1450}\ge-23$ mag) quasars from
the literature, we obtain the parameters for a double power-law fit to the QLF.
The bright-end slope $\beta$ of the QLF is well constrained to be
$\beta=-2.8\pm0.2$. Due to the small number of low-luminosity quasars, the
faint-end slope $\alpha$ and the characteristic magnitude $M_{1450}^{\ast}$ are
less well constrained, with $\alpha=-1.90_{-0.44}^{+0.58}$ and
$M^{\ast}=-25.2_{-3.8}^{+1.2}$ mag. The spatial density of luminous quasars,
parametrized as $\rho(M_{1450}<-26,z)=\rho(z=6)\,10^{k(z-6)}$, drops rapidly
from $z\sim5$ to 6, with $k=-0.72\pm0.11$. Based on our fitted QLF and assuming
an IGM clumping factor of $C=3$, we find that the observed quasar population
cannot provide enough photons to ionize the $z\sim6$ IGM at $\sim90$\%
confidence. Quasars may still provide a significant fraction of the required
photons, although much larger samples of faint quasars are needed for more
stringent constraints on the quasar contribution to reionization.
