We find that the observed log N - log S relation of X-ray clusters can be reproduced remarkably well with a certain range of values for the fluctuation amplitude $sigma_8$ and the cosmological density parameter $Omega_0$ in cold dark matter (CDM) universes. The $1sigma$ confidence limits on $sigma_8$ in the CDM models with $n=1$ and $h = 0.7$ are expressed as $(0.54 pm 0.02) Omega_0^{-0.35-0.82Omega_0+0.55Omega_0^2}$ ($lambda_0=1-Omega_0$) and $(0.54 pm 0.02) Omega_0^{-0.28-0.91Omega_0+0.68Omega_0^2}$ ($lambda_0=0$), where $n$ is the primordial spectral index, and $h$ and $lambda_0$ are the dimensionless Hubble and cosmological constants. The errors quoted above indicate the statistical ones from the observed log N - log S only, and the systematic uncertainty from our theoretical modelling of X-ray flux in the best-fit value of $sigma_8$ is about 15%. In the case of $n=1$, we find that the CDM models with $(Omega_0,lambda_0,h,sigma_8) simeq (0.3,0.7,0.7,1)$ and $(0.45, 0, 0.7, 0.8)$ simultaneously account for the cluster log N - log$S$, X-ray temperature functions, and the normalization from the COBE 4 year data. The derived values assume the observations are without systematic errors, and we discuss in details other theoretical uncertainties which may change the limits on $Omega_0$ and $sigma_8$ from the log N - log S relation. We have shown the power of this new approach which will become a strong tool as the observations attain more precision.
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