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In this work, we aimed at investigating the star formation rate of activegalactic nuclei host galaxies in the early Universe. To this end, weconstructed a sample of 149 luminous ($\rm L_{2-10keV} > 10^{44}\,erg\,s^{-1}$)X-ray AGNs at $\rm z \geq3.5$ selected in three fields with different depthsand observed areas (Chandra COSMOS Legacy survey, XMM-XXL North and eFEDS). Webuilt their spectral energy distributions (SED) using availablemulti-wavelength photometry from X-rays up to far-IR. Then, we estimated thestellar mass, M$_{*}$, and the SFR of the AGNs using the X-CIGALE SED fittingalgorithm. After applying several quality criteria, we ended up with 89 high-zsources. More than half (55\%) of the X-ray sample have spectroscopicredshifts. Based on our analysis, our high-z X-ray AGNs live in galaxies withmedian $\rm M_{*}=5.6 \times10^{10}~M_\odot$ and $\rmSFR_{*}\approx240\,M_\odot yr^{-1}$. The majority of the high-z sources($\sim89$\%) were found inside or above the main sequence (MS) of star-forminggalaxies. Estimation of the normalised SFR, $\rm SFR_{NORM}$, defined as theratio of the SFR of AGNs to the SFR of MS galaxies, showed that the SFR of AGNsis enhanched by a factor of $\sim 1.8$ compared to non-AGN star-formingsystems. Combining our results with previous studies at lower redshifts, weconfirmed that $\rm SFR_{NORM}$ does not evolve with redshift. Using thespecific BHAR (i.e., $\rm L_X$ divided by $\rm M_{*}$), $\rm \lambda _{BHAR}$,that can be used as a tracer of the Eddington ratio, we found that the bulk ofAGNs that lie inside or above the MS have higher specific accretion ratescompared to sources below the MS. Finally, we found indications that the SFR ofthe most massive AGN host galaxies ($\rm log\,(M_{*}/ M_\odot) >10^{11.5-12}$)remains roughly constant as a function of M$_*$, in agreement with the SFR ofMS star-forming galaxies.

XXL-HSC: The link between AGN activity and star formation in the Early Universe ($z\geqslant3.5$)