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

TL;DR

This paper investigates the distribution of irreducible elements in the ring of integers of number fields, providing asymptotic formulas for their maximum counts and distribution in arithmetic progressions, highlighting the role of the class group.

Contribution

It introduces new asymptotic formulas for counting irreducibles in number fields, emphasizing the influence of the class group on their distribution.

Findings

Estimated the maximum number of irreducibles dividing a given element

Counted irreducibles in arithmetic progressions within number fields

Connected distribution properties to class group structure

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 $K$.