Two Problems concerning Irreducible Elements in Rings of Integers of Number Fields

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