Comparison of the star formation of X-ray selected active galactic
  nuclei in eFEDS with star-forming galaxies
  We use $\sim 1800$ X-ray Active Galactic Nuclei (AGN) in the eROSITA Final
Equatorial-Depth Survey (eFEDS), that span over two orders of magnitude in
X-ray luminosity, $\rm L_{X,2-10keV} \approx 10^{43-45}\,ergs^{-1}$, and
compare their star-formation rate (SFR) relative to that of non-AGN
star-forming systems, at $\rm 0.5<z<1.5$. For that purpose, we compile a
reference galaxy catalogue with $\sim 17000$ sources. Both samples have the
same photometric coverage, from optical to far-infrared. We construct the
spectral energy distributions (SEDs) of all sources and fit them using the
CIGALE code, utilizing the same templates and parametric grid for both samples.
We account for the mass incompleteness and exclude quiescent systems from both
datasets. These allow us to compare the SFR of the two populations in a uniform
manner, minimising systematic effects. Based on our analysis, AGN at low and
moderate L$_X$ ($\rm L_{X,2-10keV} < 10^{44}\,ergs^{-1}$), have SFR that is
lower, or at most, equal to that of star-forming galaxies, in agreement with
previous studies. The large number of luminous X-ray AGN available in the eFEDS
dataset, enable us to expand our investigations at higher L$_X$ to test
previous, tentative results. At $\rm L_{X,2-10keV} > 10^{44.2}\,ergs^{-1}$, the
SFR of AGN appears enhanced, by $\sim 30\%$, compared to that of star-forming
sources, for systems with stellar mass, $\rm 10.5 < log\,[M_*(M_\odot)] <
11.5$, confirming indications found in previous studies. The most massive
sources $\rm log\,[M_*(M_\odot)] > 11.5$, present a flat SFR$_{norm}$-L$_X$
relation up to $\rm L_{X,2-10keV} \sim 10^{44.5}\,ergs^{-1}$, with SFR similar
to that of star-forming galaxies. However, at higher L$_X$ ($\rm L_{X,2-10keV}
\sim 10^{45}\,ergs^{-1}$), we find indications that the SFR of these massive
AGN hosts may be enhanced compared to that of non-AGN systems.
