We use the ROSAT All Sky Survey ( RASS ) to study the soft X-ray properties of a homogeneous sample of 46,420 quasars selected from the third data release of the Sloan Digital Sky Survey ( SDSS ) .
Optical luminosities , both at rest-frame 2500Å ( L _ { 2500 } ) and in [ OIII ] ( L _ { [ OIII ] } ) span more than three orders of magnitude , while redshifts range over 0.1 < z < 5.4 .
We detect 3366 quasars directly in the observed 0.1–2.4 keV band .
Sub-samples of radio-loud and radio-quiet objects ( RLQs and RQQs ) are obtained by cross-matching with the FIRST catalogue .
We study the distribution of X-ray luminosity as a function of optical luminosity , redshift and radio power using both individual detections and stacks of complete sets of similar quasars .
At every optical luminosity and redshift { log } L _ { 2 keV } is , to a good approximation , normally distributed with dispersion \sim 0.40 , at least brightwards of the median X-ray luminosity .
This median X-ray luminosity of quasars is a power law of optical luminosity with index \sim 0.53 for L _ { 2500 } and \sim 0.30 for L _ { [ OIII ] } .
RLQs are systematically brighter than RQQs by about a factor of 2 at given optical luminosity .
The zero-points of these relations increase systematically with redshift , possibly in different ways for RLQs and RQQs .
Evolution is particularly strong at low redshift and if the optical luminosity is characterised by L _ { [ OIII ] } .
At low redshift and at given L _ { [ OIII ] } the soft X-ray emission from type II AGN is more than 100 times weaker than that from type I AGN .