Coset topologies on [formula omitted] and arithmetic applications

Abstract

We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on Z, obtained from closed sets of the profinite completion Zˆ. We also discuss various applications to number theory.