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An arithmetic function $f$ is called a sieve function of range $Q$, if it isthe 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 autocorrelationof $f$ by using a famous theorem on bilinear forms of Kloosterman fractions byDuke, Friedlander and Iwaniec. In particular, for such correlations we obtainnon-trivial asymptotic formul\ae\ that are actually unreachable by the standardapproach of the distribution of $f$ in the arithmetic progressions. Moreover,we apply our asymptotic formul\ae\ to obtain new bounds for the so-calledSelberg integral and symmetry integral of $f$, which are basic tools for thestudy of the distribution of $f$ in short intervals.

On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals, III