In this paper, we revisit the constraints obtained by several authors (Reichart et al. 1999; Eke et al. 1998; Henry 2000) on the estimated values of $\Omega_{\rm m}$, $n$ and $\sigma_8$ in the light of recent theoretical developments: 1) new theoretical mass functions (Sheth & Tormen 1999, Sheth, Mo & Tormen 2001, Del Popolo 2002b); 2) a more accurate mass-temperature relation, also determined for arbitrary $\Omega_{\rm m}$ and $\Omega_{\rm \Lambda}$ (Del Popolo 2002a).

Firstly, using the quoted improvements, we re-derive an expression for the X-ray Luminosity Function (XLF), similarly to Reichart et al. (1999), and then we get some constraints to $\Omega_{\rm m}$ and $n$, by using the ROSAT BCS and EMSS samples and maximum-likelihood analysis. Then we re-derive the X-ray Temperature Function (XTF), similarly to Henry (2000), re-obtaining the constraints on $\Omega_{\rm m}$, $n$, $\sigma_8$. Both in the case of the XLF and XTF, the changes in the mass function and M-T relation produces an increase in $\Omega_{\rm m}$ of $\simeq 20\%$ and similar results in $\sigma_8$ and $n$.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html