Abstract: Models of galaxy evolution assume some connection between the AGN and star formation activity in galaxies. We use the multi-wavelengthinformation of the CDFS to assess this issue. We select the AGNs from the 3 Ms XMM-Newton survey and measure the star-formation rates oftheir hosts using data that probe rest-frame wavelengths longward of 20 ?m, predominantly from deep 100 ?m and 160 ?m Herschel observations, but also from Spitzer-MIPS-70 ?m. Star-formation rates are obtained from spectral energy distribution fits, identifying and subtracting an AGN component. Our sample consists of sources in the z ? 0.5?4 redshift range, with star-formation rates SFR ? 101?103 M yr?1 and stellar masses M ? 1010?1011.5 M. We divide the star-formation rates by the stellar masses of the hosts to derive specific star-formation rates (sSFR) and find evidence for a positive correlation between the AGN activity (proxied by the X-ray luminosity) and the sSFR for the most active systems with X-ray luminosities exceeding Lx  1043 erg s?1 and redshifts z 1. We do not find evidence for such a correlation for lower luminosity systems or those at lower redshifts, consistent with previous studies. We do not find any correlation between the SFR (or the sSFR) and the X-ray absorption derived from high-quality XMM-Newton spectra either, showing that the absorption is likely to be linked to the nuclear region rather than the host, while the star-formation is not nuclear. Comparing the sSFR of the hosts to the characteristic sSFR of star-forming galaxies at the same redshift (the so-called ?main sequence?) we find that the AGNs reside mostly in main-sequence and starburst hosts, reflecting the AGN-sSFR connection; however the infrared selection might bias this result. Limiting our analysis to the highest X-ray luminosity AGNs (X-ray QSOs with Lx > 1044 erg s?1), we find that the highest-redshift QSOs (with z 2) reside predominantly in starburst hosts, with an average sSFR more than double that of the ?main sequence?, and we find a few cases of QSOs at z ? 1.5 with specific star-formation rates compatible with the mainsequence, or even in the ?quiescent? region. Finally, we test the reliability of the colour?magnitude diagram (plotting the rest-frame optical colours against the stellar mass) in assessing host properties, and find a significant correlation between rest-frame colour (without any correction for AGN contribution or dust extinction) and sSFR excess relative to the ?main sequence? at a given redshift. This means that the most ?starbursty? objects have the bluest rest-frame colours.

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