The Number of Integers n in an Interval such that a ℓ n+b ℓ , ≤ ℓ ≤ m , are Norms of Ideals of a Number Field

Let Formula: see text be a number field, Formula: see text a real number, and Formula: see text integers. Let Formula: see text and Formula: see text be sequences of integers such that Formula: see text and Formula: see text for all 1 Formula: see text. We give an upper bound for the number of integers Formula: see text for which Formula: see text, for each Formula: see text, is the norm of an ideal of the ring of integers of Formula: see text. This result extends to arbitrary number fields the theorem of Nowak 6, proved for normal extensions K/Formula: see text, which itself generalised earlier results of Cochrane and Dressler 5 and of Rieger 9.