  Temperature and luminosity functions of X-ray clusters are computed
semi-analytically, combining a simple model for the cluster gas properties with
the distribution functions of halo formation epochs proposed by Lacey & Cole
(1993) and Kitayama & Suto (1996). In contrast to several previous approaches
which apply the Press--Schechter mass function in a straightforward manner, our
method can explicitly take into account the temperature and luminosity
evolution of clusters. In order to make quantitative predictions in a specific
cosmological context, we adopt cold dark matter (CDM) universes.
  Assuming the baryon density parameter $\Omega_{\rm B}=0.0125 h^{-2}$ ($h$ is
the Hubble constant in units of 100 km$\cdot$sec$^{-1}\cdot$Mpc$^{-1}$) and the
{\it COBE} normalization of matter fluctuations, temperature and luminosity
functions of X-ray clusters depend sensitively on the density parameter
$\Omega_0$. Allowing for several uncertainties in observational data as well as
in our simplified assumptions, we conclude that $\Omega_0 \sim 0.2-0.5$ and
$h\sim 0.7$ CDM models with/without the cosmological constant reproduce
simultaneously the observed temperature and luminosity functions of X-ray
clusters at redshift $z\sim0$.
