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

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.