An application of algebraic sieve theory
A well-known result from sieve theory [see \textit{H. Halberstam} and \textit{H.-E. Richert}, Sieve methods (London Math. Soc. Monographs 4, Academic Press, London) (1974; Zbl 0298.10026)] says that if \(f(X)\in \mathbb{Z}[X]\) is an irreducible polynomial of degree \(g\), with no fixed prime factor, then \(f(p)\) is \((2g+ 1)\)-almost-prime for \(\gg x(\log x)^{-2}\) primes \(p\leq x\). The goal of the present paper is to generalize this result to algebraic number fields. The paper works with a fixed totally real field \(K\) of degree \(n\), and a polynomial \(f(X)\in \mathbb{Z}_K[X]\), where \(\mathbb{Z}_K\) is the ring of integers of \(K\). One then considers parameters \(P_1,\dots, P_n\geq 2\) and examines \(f(\omega)\) where \(\omega\) is a prime in \(\mathbb{Z}_K\) and \(0< \omega^{(i)}\leq P_i\) \((1\leq i\leq n)\) for the various real embeddings of \(K\). The number of \((2g+1)\)-almost-prime values of \(f(\omega)\) is then shown to be \(\gg P(\log P)^{-2}\), where \(P= P_1\cdots P_n\). The reader should note that this remains true even in the extreme case in which some of the \(P_i\) are \(O(1)\). The proof builds on results for the linear sieve over number fields, developed previously by the author [Acta Arith. 41, 223--254 (1982; Zbl 0432.12009)].
scientific article; zbMATH DE number 2004958