Arithmetic functions and their coprimality

Abstract

Let $D\ge 3$ be an odd integer and $\ell\ge -1$ be a non zero integer such that gcd$(\ell,D)=1$. Let $f,g:\mathbb{N} \to \mathbb{N}$ be multiplicative functions such that $f(p)=D$ and $g(p)=p+\ell$ for each prime $p$. We estimate the number of positive integers $n\le x$ such that gcd$(f(n),g(n))=1$. If $D$ is a prime larger than 3, we also examine the size of the number of positive integers $n\le x$ for which $\mbox{gcd}(g(n),f(n-1))=1$.