  We determine the number counts and z=0-5 luminosity function for a
well-defined, homogeneous sample of quasars from the Sloan Digital Sky Survey
(SDSS). We conservatively define the most uniform statistical sample possible,
consisting of 15,343 quasars within an effective area of 1622 deg^2 that was
derived from a parent sample of 46,420 spectroscopically confirmed broad-line
quasars in the 5282 deg^2 of imaging data from SDSS Data Release Three. The
sample extends from i=15 to i=19.1 at z<3 and to i=20.2 for z>3. The number
counts and luminosity function agree well with the results of the 2dF QSO
Survey, but the SDSS data probe to much higher redshifts than does the 2dF
sample. The number density of luminous quasars peaks between redshifts 2 and 3,
although uncertainties in the selection function in this range do not allow us
to determine the peak redshift more precisely. Our best fit model has a flatter
bright end slope at high redshift than at low redshift. For z<2.4 the data are
best fit by a redshift-independent slope of beta = -3.1 (Phi(L) propto L^beta).
Above z=2.4 the slope flattens with redshift to beta=-2.37 at z=5. This slope
change, which is significant at a >5-sigma level, must be accounted for in
models of the evolution of accretion onto supermassive black holes.
