Sieve functions in arithmetic bands

An arithmetic function $f$ is a sieve function of range $Q$, if its Eratosthenes transform $g=fastmu$ is supported in $[1,Q]capN$, where $g(q)ll_{arepsilon} q^{arepsilon}$, $orallarepsilon>0$. Here, we study the distribution of $f$ over the so-called short arithmetic bands $igcup_{1le ale H}{nin(N,2N]: nequiv a, (mod,q)}$, 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.