We present the results from a survey of i-dropout objects selected from ~1550 deg^2 of multicolor imaging data from the Sloan Digital Sky Survey, to search for luminous quasars at z>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, at z=5.82, 5.99 and 6.28, respectively. Their spectra show strong and broad Ly alpha+NV emission lines, and very strong Ly alpha absorption, with a mean continuum decrement D_A > 0.90. The ARC 3.5m spectrum of the z=6.28 quasar shows that over a range of 300 A 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, and suggesting a tentative detection of the complete Gunn-Peterson trough. The existence of strong metal lines suggests early chemical enrichment in the quasar enviornment. The three new objects, together with the previously published z=5.8 quasar form a complete color-selected flux-limited sample at z>5.8. We estimate that at $z=6$, the comoving density of luminous quasars at M_1450 < -26.89 (h=0.5, Omega=1)is 1.1x10^-9 Mpc^-3. This is a factor of ~2 lower than that at z~5, and is consistent with an extrapolation of the observed quasar evolution at low-z. We discuss the contribution of quasars to the ionizing background at z~6. The luminous quasars discussed in the paper have central black hole masses of several times 10^9 M_sun by the Eddington argument. Their observed space density provides a sensitive test of models of quasar and galaxy formation at high redshift. (Abridged)
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