A survey of z>5.8 quasars in the Sloan Digital Sky Survey.: I.: Discovery of three new quasars and the spatial density of luminous quasars at z∼6

We present the results from a survey of i-dropout objects selected from similar to 1550 deg(2) of multicolor imaging data from the Sloan Digital Sky Survey to search for luminous quasars at z greater than or similar to 5.8. Objects with i*-z* > 2.2 and z* < 20.2 are selected, and follow-up J-band photometry is used to separate L- and T-type cool dwarfs from high-redshift quasars. We describe the discovery of three new quasars, SDSSp J083643.85+005453.3 (z = 5.82), J130608.26+035626.3 (z = 5.99), and J103027.10+052455.0 (z = 6.28). The quasar SDSSp J083643.85+005453.3 is a radio source with flux of 1.1 mJy at 20 cm. The spectra of all three quasars show strong and broad Ly<alpha>+N v emission lines and very strong Ly alpha forest absorption, with a mean continuum decrement D(A) > 0.90. The ARC 3.5 m spectrum of SDSSp J103027.10+052455.0 shows that over a range of similar to 300 Angstrom immediately blueward of the Ly alpha emission, the average transmitted flux is only 0.003 +/- 0.020 times that of the continuum level, consistent with zero flux over a similar to 300 Angstrom range of the Ly alpha forest region and suggesting a tentative detection of the complete Gunn-Peterson trough. The existence of strong metal lines in the quasar spectra suggests early metal enrichment in the quasar environment. The three new objects, together with the previously published z = 5.8 quasar SDSSp J104433.04-012502.2, form a complete color-selected flux-limited sample at z greater than or similar to 5.8. We estimate the selection function of this sample, taking into account the estimated variations in the quasar spectral energy distribution, as well as observational photometric errors. We find that at z = 6, the comoving density of luminous quasars at M(1450) < -26.8 (H(0) = 50 km s(-1) Mpc(-1), <Omega> = 1) is 1.1 x 10(-9) Mpc(-3). This is a factor of similar to2 lower than that at z similar to 5 and is consistent with an extrapolation of the observed quasar evolution at z < 5. Using the current sample, we discuss the constraint on the shape of the quasar luminosity function and the implications for the contribution of quasars to the ionizing background at z <similar to> 6. The luminous quasars discussed in the paper have central black hole masses of several times 10(9) M. by the Eddington argument, with likely dark halo masses on the order of 10(13) M. Their observed space density provides a sensitive test of models of quasar and galaxy formation at high redshift.