  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
2500\AA ($L_{2500}$) and in [OIII] ($L_{[\rm{OIII}]}$) 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
$L_{[\rm{OIII}]}$. 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 $L_{[\rm{OIII}]}$. At low redshift and at given
$L_{[\rm{OIII}]}$ the soft X-ray emission from type II AGN is more than 100
times weaker than that from type I AGN.
