Sieve functions in arithmetic bands, II

TL;DR

This paper investigates the distribution of sieve functions within short arithmetic bands, analyzing their behavior and optimality of results, extending previous work on the subject.

Contribution

It advances the understanding of sieve functions in arithmetic bands, focusing on their distribution and the optimality of existing bounds.

Findings

Distribution of sieve functions in short arithmetic bands analyzed

Optimality of certain bounds discussed

Extension of previous results on sieve functions

Abstract

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 $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). We continue our study of the distribution of such functions over short $arithmetic$ $bands$, $n\equiv ar+b\, (\bmod\,q)$, 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.