  Using a sample of 228 optically selected Active Galactic Nuclei (AGNs) in the
0.01-6.3 redshift range with a high fraction of X-ray detections (81-86%), we
study the relation between rest-frame UV and soft X-ray emission and its
evolution with cosmic time. The majority of the AGNs in our sample (155
objects) have been selected from the Sloan Digital Sky Survey (SDSS) in an
unbiased way, rendering the sample results representative of all SDSS AGNs. The
addition of two heterogeneous samples of 36 high-redshift and 37 low-redshift
AGNs further supports and extends our conclusions. We confirm that the X-ray
emission from AGNs is correlated with their UV emission, and that the ratio of
the monochromatic luminosity emitted at 2keV compared to 2500A decreases with
increasing luminosity (a_ox=-0.136 l_uv+2.616, where l_uv is in log units), but
does not change with cosmic time. These results apply to intrinsic AGN
emission, as we correct or control for the effects of the host galaxy, UV/X-ray
absorption, and any X-ray emission associated with radio emission in AGNs. We
investigate a variety of systematic errors and can thereby state with
confidence that (1) the a_ox-l_uv anti-correlation is real and not a result of
accumulated systematic errors and (2) any a_ox dependence on redshift is
negligible in comparison. We provide the best quantification of the a_ox-l_uv
relation to date for normal radio-quiet AGNs; this should be of utility for
researchers pursuing a variety of studies.
