  Utilizing 21 new Chandra observations as well as archival Chandra, ROSAT, and
XMM-Newton data, we study the X-ray properties of a representative sample of 59
of the most optically luminous quasars in the Universe (M_i~~-29.3 to -30.2)
spanning a redshift range of z~~1.5-4.5. Our full sample consists of 32 quasars
from the Sloan Digital Sky Survey (SDSS) Data Release 3 (DR3) quasar catalog,
two additional objects in the DR3 area that were missed by the SDSS selection
criteria, and 25 comparably luminous quasars at z>~4. This is the largest X-ray
study of such luminous quasars to date. By jointly fitting the X-ray spectra of
our sample quasars, excluding radio-loud and broad absorption line (BAL)
objects, we find a mean X-ray power-law photon index of
Gamma=1.92^{+0.09}_{-0.08} and constrain any neutral intrinsic absorbing
material to have a mean column density of N_H<~2x10^{21} cm^{-2}. We find,
consistent with other studies, that Gamma does not change with redshift, and we
constrain the amount of allowed Gamma evolution for the most-luminous quasars.
Our sample, excluding radio-loud and BAL quasars, has a mean X-ray-to-optical
spectral slope of a_ox=-1.80+/-0.02, as well as no significant evolution of
a_ox with redshift. We also comment upon the X-ray properties of a number of
notable quasars, including an X-ray weak quasar with several strong narrow
absorption-line systems, a mildly radio-loud BAL quasar, and a well-studied
gravitationally lensed quasar.
