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The 2dF-SDSS LRG and QSO Survey: The z<2.1 Quasar Luminosity Function from 5645 Quasars to g=21.85

We have used the 2dF instrument on the AAT to obtain redshifts of a sample of z<3, 18.0<g<21.85 quasars selected from SDSS imaging. These data are part of a larger joint programme: the 2dF-SDSS LRG and QSO Survey (2SLAQ). We describe the quasar selection algorithm and present the resulting luminosity function of 5645 quasars in 105.7 deg^2. The bright end number counts and luminosity function agree well with determinations from the 2dF QSO Redshift Survey (2QZ) data to g\sim20.2. However, at the faint end the 2SLAQ number counts and luminosity function are steeper than the final 2QZ results from Croom et al. (2004), but are consistent with the preliminary 2QZ results from Boyle et al. (2000). Using the functional form adopted for the 2QZ analysis, we find a faint end slope of beta=-1.78+/-0.03 if we allow all of the parameters to vary and beta=-1.45+/-0.03 if we allow only the faint end slope and normalization to vary. Our maximum likelihood fit to the data yields 32% more quasars than the final 2QZ parameterization, but is not inconsistent with other g>21 deep surveys. The 2SLAQ data exhibit no well defined ``break'' but do clearly flatten with increasing magnitude. The shape of the quasar luminosity function derived from 2SLAQ is in good agreement with that derived from type I quasars found in hard X-ray surveys. [Abridged]