  An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if
its Eratosthenes transform $g=f\ast\mu$ is supported in $[1,Q]\cap\N$, where
$g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we
study the distribution of $f$ over short {\it arithmetic bands} $\cup_{1\le
a\le H}\{n\in(N,2N]: n\equiv a\, (\bmod\,q)\}$, with $H=o(N)$, and give
applications to both the correlations and to the so-called weighted Selberg
integrals of $f$, on which we have concentrated our recent research.
