  An arithmetic function $f$ is called a sieve function of range $Q$, if it is
the convolution product of the constantly $1$ function and $g$ such that
$g(q)\ll_{\varepsilon} q^{\varepsilon}$, $\forall\varepsilon>0$, for $q\leq Q$,
and $g(q)=0$ for $q>Q$. Here we establish a new result on the autocorrelation
of $f$ by using a famous theorem on bilinear forms of Kloosterman fractions by
Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain
non-trivial asymptotic formul\ae\ that are actually unreachable by the standard
approach of the distribution of $f$ in the arithmetic progressions. Moreover,
we apply our asymptotic formul\ae\ to obtain new bounds for the so-called
Selberg integral and symmetry integral of $f$, which are basic tools for the
study of the distribution of $f$ in short intervals.
