  We prove a non-trivial result for the,say,modified Selberg integral
$\modSel_3(N,h)$, of the divisor function $d_3(n):= \sum_{a}\sum_{b}\sum_{c,
abc=n}1$; this integral is a slight modification of the corresponding Selberg
integral, that gives the expected value of the function in short intervals. We
get, in fact, $\modSel_3(N,h)\ll Nh^2L^2$, where $L:=\log N$; furthermore, as a
byproduct, we obtain indications on the way in which it may be proved the weak
sixth moment of the Riemann zeta function.(This was OLD abstract)
