We investigate the relation between AGN and star formation (SF) activity at$0.5 < z < 3$ by analyzing 898 galaxies with X-ray luminous AGN ($L_X >10^{44}$ erg s$^{-1}$) and a large comparison sample of $\sim 320,000$ galaxieswithout X-ray luminous AGN. Our samples are selected from a large (11.8deg$^2$) area in Stripe 82 that has multi-wavelength (X-ray to far-IR) data.The enormous comoving volume ($\sim 0.3$ Gpc$^3$) at $0.5 < z < 3$ minimizesthe effects of cosmic variance and captures a large number of massive galaxies($\sim 30,000$ galaxies with $M_* > 10^{11} \ M_{\odot}$) and X-ray luminousAGN. While many galaxy studies discard AGN hosts, we fit the SED of galaxieswith and without X-ray luminous AGN with Code Investigating GALaxy Emission(CIGALE) and include AGN emission templates. We find that without thisinclusion, stellar masses and star formation rates (SFRs) in AGN host galaxiescan be overestimated, on average, by factors of up to $\sim 5$ and $\sim 10$,respectively. The average SFR of galaxies with X-ray luminous AGN is higher bya factor of $\sim 3$ to $10$ compared to galaxies without X-ray luminous AGN atfixed stellar mass and redshift, suggesting that high SFRs and high AGN X-rayluminosities may be fueled by common mechanisms. The vast majority ($> 95 \%$)of galaxies with X-ray luminous AGN at $z=0.5-3$ do not show quenched SF: thissuggests that if AGN feedback quenches SF, the associated quenching processtakes a significant time to act and the quenched phase sets in after the highlyluminous phases of AGN activity.

We investigate the relationship between black hole accretion and starformation in a sample of 453 $z\approx0.3$ type 1 active galactic nuclei(AGNs). We use available CO observations to demonstrate that the combination ofnebular dust extinction and metallicity provides reliable estimates of themolecular gas mass even for the host galaxies of type 1 AGNs. Consistent withother similar but significantly smaller samples, we reaffirm the notion thatpowerful AGNs have comparable gas content as nearby star-forming galaxies andthat AGN feedback does not deplete the host of cold gas instantaneously. Wedemonstrate that while the strong correlation between star formation rate andblack hole accretion rate is in part driven by the mutual dependence of theseparameters on molecular gas mass, the star formation rate and black holeaccretion rate are still weakly correlated after removing the dependence ofstar formation rate on molecular gas mass. This, together with a positivecorrelation between star formation efficiency and black hole accretion rate,may be interpreted as evidence for positive AGN feedback.