  We report the results of our Chandra observations of six QSOs at $z\sim 3$ from the Palomer Transit Grism Survey. Our primary goal is to investigate the possible systematic change of $\alpha_{ox}$ between $z>4$ and $z\sim 3$, between which a rapid rise of luminous QSO number density with cosmic time is observed. The summed spectrum showed a power-law spectrum with photon index of $\Gamma \approx 1.9$, which is similar to other unabsorbed AGNs. Combining our $z\sim 3$ QSOs with X-ray observations of QSOs at $z>4$ from literaure/archive, we find a correlation of $\alpha_{\rm ox}$ with optical luminosity. This is consistent with the fact that the luminosity function slope of the luminous end of the X-ray selected QSOs is steeper than that of optically-selected QSOs. We discuss an upper limit to the redshift dependence of $\alpha_{ox}$ using a Monte-Carlo simulation. Within the current statistical errors including the derived limits on the redshift dependence of $\alpha_{\rm ox}$, we found that the behaviors of the X-ray and optically-selected QSO number densities are consistent with each other. 
Chandra Observations of Six QSOs at z $\approx$ 3