tokens: unobscured sources, to dex for faint and highly obscured sources. The average value of the error is dex, with standard deviation . We show the average errors with a black cross in the left panel of Fig. 4 , while the specific value for each source is used in the following analysis. In order to look for intrinsic correlations between these two quantities, one possibility is to compute the partial Spearman rank correlation between two variables in the presence of a third, and to assess the statistical significance of such correlation (e.g., Macklin 1982 ). To derive the correlation coefficient between and , conditioned by the distance, (, , ), we evaluate the Spearman coefficient related to each pair of parameters and then combine them according to the expression (Conover 1980), which returns the partial correlation between and , corrected for the dependency on . The resulting is 0.15, and the associated confidence level, in terms of standard deviations, that the first two variables are correlated, independently of the influence of the third, is , following Eq. (6) of Macklin ( 1982 ). Therefore, the two quantities appear to be significantly correlated, after the effect of redshift on both of them is taken into account. 3.2. Redshift bins The second approach, often used in the literature, is to define redshift bins that are as narrow as possible to minimize the distance effect, and to look for correlations between the two quantities. Thanks to the large sample collected in this work, we can divide the sample into five redshift bins. For every redshift bin, a large distribution in both luminosities can be observed, with the typical luminosity increasing with redshift (Fig. 4 ). Most of the observational works mentioned in Sect. 1 looked for the distribution of average in bins, or average in bins (but see Gruppioni et al. 2016 ). Both hydrodynamical simulations (e.g., Volonteri et al. 2015 a,b) and semi-analytic models (e.g., Neistein Netzer 2014 ) show that, in the - plane, there may be the superimposition of a weak correlation for the bulk of the population, and a strong correlation only for the most extreme merger phases, corresponding to the highest and . If the underlying distribution shows such a complex shape, the results of the two approaches (average in bins or average in bins) may be very different. In Fig. 5 (left) we show the result of plotting average in bin of (both in log scale) in five redshift bins. As can be seen, there is no correlation at all and, as expected, there are no sources below the relation computed for a pure AGN template in Mullaney et al. ( 2011 ), similar to the one used in Netzer et al. (2009). Following this approach, we are therefore able to reproduce the results of Shao et al. ( 2010 ), Rosario et al. ( 2012 ), and others that claim no correlation between AGN activity and SF over several orders of magnitude in luminosity. On the other hand, computing average in bins (in log scale) from the same bivariate
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