On the Symmetry Integral

We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N<x\le 2N$) of the "symmetry" \thinspace of, say, the arithmetic function $f:=g\ast \1$, where $g:\N \to \R$ is such that $\forall \epsilon>0$ we have $g(n)\ll_{\epsilon} n^{\epsilon}$, and supported in $[1,Q]$, with $Q\ll N$ (so, the exponent of $Q$ relative to $N$, say the level $\lambda:=(\log Q)/(\log N)$ is $\lambda < 1$), where the symmetry sum weights the $f-$values in (almost all, i.e. all but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$ (with positive/negative sign at the right/left of $x$), with mild restrictions on $h$ (say, $h\to \infty$ and $h=o(\sqrt N)$, as $N\to \infty$).