Counting carefree couples

Abstract

A pair of natural numbers (a,b) such that a is both squarefree and coprime to b is called a carefree couple. A result conjectured by Manfred Schroeder (in his book `Number theory in science and communication') on carefree couples and a variant of it are established using standard arguments from elementary analytic number theory. Also a related conjecture of Schroeder on triples of integers that are pairwise coprime is proved.Comment: Updated version of 2005 update of 2000 version. Improved and expanded presentation. In estimate (2) now only a weaker error term than before is obtaine