  We deeply appreciate the papers of Ivi\'c on the links between the $2k-$th
moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function.
More specifically, both the one bounding the $2k-$th moment with a simple
average of correlations of the $d_k$ (Palanga 1996 Conference Proceedings) and
the more recent (arXiv:0708.1601v2 to appear on JTNB), which bounds the Selberg
integral of $d_k$ applying the $2k-$th moment of the zeta. Building on the
former paper, we apply an elementary approach (based on arithmetic averages) in
order to get information on the reverse link to the second work.
