Two problems concerning irreducible elements in rings of integers of number fields

Abstract

Let $K$ be a number field with ring of integers ${\mathbb{Z}}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in ${\mathbb{Z}}_K$. First, we estimate the maximum number of nonassociated irreducibles dividing a nonzero element of ${\mathbb{Z}}_K$ of norm not exceeding $x$ (in absolute value), as $x\to \infty$. Second, we count the number of irreducible elements of ${\mathbb{Z}}_K$ of norm not exceeding $x$ lying in a given arithmetic progression (again, as $x\to \infty$). When $K=\mathbb{Q}$, both results are classical; a new feature in the general case is the influence of combinatorial properties of the class group of $K$.