  We study the arithmetic (real) function f=g*1, with g "essentially bounded"
and supported over the integers of [1,Q]. In particular, we obtain non-trivial
bounds, through f "correlations", for the "Selberg integral" and the "symmetry
integral" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the
"classical" level, up to level of distribution, say, lambda=log Q/log N < 2/3
(for enough large h). This time we don't apply Large Sieve inequality, as in
our paper [C-S]. Precisely, our method is completely elementary.
