  From the Press-Schechter mass function and the empirical X-ray cluster
luminosity-temperature (L-T) relation, we construct an X-ray cluster luminosity
function that can be applied to the growing number of high-redshift, X-ray
cluster luminosity catalogs to constrain cosmological parameters. In this
paper, we apply this luminosity function to the Einstein Medium Sensitivity
Survey (EMSS) and the ROSAT Brightest Cluster Sample (BCS) luminosity function
to constrain the value of Omega_m. In the case of the EMSS, we find a factor of
4 - 5 fewer X-ray clusters at redshifts above z = 0.4 than below this redshift
at luminosities above L_X = 7 x 10^44 erg s^-1 (0.3 - 3.5 keV), which suggests
that the X-ray cluster luminosity function has evolved above L_*. At lower
luminosities, this luminosity function evolves only minimally, if at all. Using
Bayesian inference, we find that the degree of evolution at high luminosities
suggests that Omega_m = 0.96(+0.36,-0.32), given the best-fit L-T relation of
Reichart, Castander, & Nichol (1998). When we account for the uncertainty in
how the empirical L-T relation evolves with redshift, we find that Omega_m is
approximately 1.0 +/- 0.4. However, it is unclear to what degree systematic
effects may affect this and similarly obtained results.
