Density of subsets of squarefree elements in certain Dedekind domains

Abstract

We consider polynomial rings over finite fields and rings of integers of imaginary quadratic fields $\mathbb{Q}(\sqrt{-D})$. In this paper, we formulate the density of squarefree elements divisible by all elements of $T$ but by none of $P$, where $T$ and $P$ are subsets of squarefree elements and $T$ is finite. We also define Mersenne irreducibles in order to estimate the density of squarefree elements divisible by none of $P$.