  The present paper is a dissertation on the possible consequences of a
conjectural bound for the so-called \thinspace modified Selberg integral of the
divisor function $d_3$, i.e. a discrete version of the classical Selberg
integral, where $d_3(n)=\sum_{abc=n}1$ is attached to the Cesaro weight
$1-|n-x|/H$ in the short interval $|n-x|\le H$. Mainly, an immediate
consequence is a non-trivial bound for the Selberg integral of $d_3$, improving
recent results of Ivi\'c based on the standard approach through the moments of
the Riemann zeta function on the critical line. We proceed instead with
elementary arguments, by first applying the "elementary Dispersion Method" in
order to establish a link between "weighted Selberg integrals" \thinspace of
any arithmetic function $f$ and averages of correlations of $f$ in short
intervals. Moreover, we provide a conditional generalization of our results to
the analogous problem on the divisor function $d_k$ for any $k\ge 3$. Further,
some remarkable consequences on the $2k-$th moments of the Riemann zeta
function are discussed. Finally, we also discuss the essential properties that
a general function $f$ should satisfy so that the estimation of its Selberg
integrals could be approachable by our method.
