TWO PROBLEMS CONCERNING IRREDUCIBLE ELEMENTS IN RINGS OF INTEGERS OF NUMBER FIELDS

Let K be a number field with ring of integers Z(K). We prove two asymptotic formulas connected with the distribution of irreducible elements in Z(K). First, we estimate the maximum number of nonassociated irreducibles dividing a nonzero element of Z(K) of norm not exceeding x (in absolute value), as x -> infinity. Second, we count the number of irreducible elements of Z(K) of norm not exceeding x lying in a given arithmetic progression (again, as x -> infinity). 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.