Sieve functions in arithmetic bands

An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g=f\ast \mu$ is supported in $[1,Q]\cap \mathbb{N}$, where $g(q){\ll }_{\epsilon }{q}^{\epsilon }$ ($\mathrm{\forall }\epsilon >0$). 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.