The Extremely Luminous Quasar Survey (ELQS) in the Sloan Digital Sky
  Survey footprint. II. The North Galactic Cap Sample
  We present the North Galactic Cap sample of the Extremely Luminous Quasar
Survey (ELQS-N), which targets quasars with $M_{1450}<-27$ at $2.8 \leq z < 5$
in an area of $\sim7600\,\rm{deg}^2$ of the Sloan Digital Sky Survey (SDSS)
footprint with $90\text{\textdegree}<\rm{RA}<270\text{\textdegree}$. Based on a
near-infrared/infrared \textit{JKW2} color cut, the ELQS selection efficiently
uses random forest methods to classify quasars and to estimate photometric
redshifts; this scheme overcomes some of the difficulties of pure optical
quasar selection at $z\approx3$. As a result, we retain a completeness of
$>70\%$ over $z\sim3.0-5.0$ at $m_{i}\lesssim17.5$, limited toward fainter
magnitudes by the depth of the Two Micron All Sky Survey (2MASS). The presented
quasar catalog consists of a total of 270 objects, of which 39 are newly
identified in this work with spectroscopy obtained at the Vatican Advanced
Technology Telescope and the MMT $6.5\,\rm{m}$ telescope. In addition to the
high completeness, which allowed us to discover new quasars in the already
well-surveyed SDSS North Galactic Cap, the efficiency of our selection is
relatively high at $\sim79\%$. Using 120 objects of this quasar sample we are
able to extend the previously measured optical quasar luminosity function (QLF)
by one magnitude toward the bright end at $2.8 \leq z \leq 4.5$. A first
analysis of the QLF suggests a relatively steep bright-end slope of
$\beta\approx-4$ for this sample. This result contrasts with previous results
in the same redshift range, which find a much flatter slope around
$\beta\sim-2.5$, but agrees with recent measurements of the bright-end slope at
lower and higher redshifts. Our results constrain the bright-end slope at
$z=2.8-4.5$ to $\beta<-2.94$ with a 99\% confidence.
