The Faint End of the Quasar Luminosity Function at z~4: Implications for
  Ionization of the Intergalactic Medium and Cosmic Downsizing
  We present an updated determination of the z ~ 4 QSO luminosity function
(QLF), improving the quality of the determination of the faint end of the QLF
presented in Glikman et al. (2010). We have observed an additional 43
candidates from our survey sample, yielding one additional QSO at z = 4.23 and
increasing the completeness of our spectroscopic follow-up to 48% for
candidates brighter than R = 24 over our survey area of 3.76 deg2. We study the
effect of using K-corrections to compute the rest-frame absolute magnitude at
1450A compared with measuring M1450 directly from the object spectra. We find a
luminosity-dependent bias: template-based K-corrections overestimate the
luminosity of low-luminosity QSOs, likely due to their reliance on templates
derived from higher luminosity QSOs. Combining our sample with bright quasars
from the Sloan Digital Sky Survey and using spectrum-based M1450 for all the
quasars, we fit a double-power-law to the binned QLF. Our best fit has a
bright-end slope, {\alpha} = 3.3\pm0.2, and faint-end slope, {\beta} =
1.6(+0.8/-0.6). Our new data revise the faint-end slope of the QLF down to
flatter values similar to those measured at z ~ 3. The break luminosity, though
poorly constrained, is at M* = -24.1(+0.7/-1.9), approximately 1 - 1.5 mag
fainter than at z ~ 3. This QLF implies that QSOs account for about half the
radiation needed to ionize the IGM at these redshifts.
