An application of algebraic sieve theory

Abstract

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieri’s theorem on primes in arithmetic progressions.