Sieve functions in arithmetic bands, II

An arithmetic function $f$ is called a $sieve$ $function$ of $range$ $Q$ if its Eratosthenes transform $g=f\ast \mu$ has support in $[1,Q]$, where $g(q){\ll }_{\epsilon }{q}^{\epsilon }$ ($\mathrm{\forall }\epsilon >0$). We continue our study of the distribution of such functions over short $arithmetic$ $bands$, $n\equiv ar+b(q)$, with $1\le a\le H=o(N)$ and $r,b$ integers such that g.c.d.$(r,q)=1$. In particular, we discuss the optimality of some results.