  We analyze the ROSAT Deep Cluster Survey (RDCS) to derive cosmological
constraints from the evolution of the cluster X-ray luminosity distribution.
The sample contains 103 galaxy clusters out to z=0.85 and flux-limit Flim=3
10^{-14} cgs (RDCS-3) in the [0.5-2.0] keV energy band, with a high-z extension
containing four clusters at 0.90<z<1.26 and F>1 10^{-14} cgs (RDCS-1). Model
predictions for the cluster mass function are converted into the X-ray
luminosity function in two steps. First we convert mass into intra-cluster gas
temperature by assuming hydrostatic equilibrium. Then temperature is converted
into X-ray luminosity by using the most recent data on the Lx-T relation for
nearby and distant clusters. These include the Chandra data for seven distant
clusters at 0.57<z<1.27. From RDCS-3 we find \Omega_m=0.35+/-0.12 and
\sigma_8=0.66+/-0.06 for a spatially flat Universe with cosmological constant,
with no significant constraint on \Gamma . Even accounting for theoretical and
observational uncertainties in the mass/X-ray luminosity conversion, an
Einstein-de-Sitter model is always excluded at far more than the 3sigma level.
