An interval result for the number field \(\psi{}(x,y)\) function
For a given number field \(K\), define \(\psi_ K(x,y)\) to be the number of ideals in the ring of integers of \(K\) with norm \(\leq x\),and free of prime ideal factors with norm \(>y\). This function has been well-studied for \(K=\mathbb{Q}\); and recent authors have shown that analogous methods carry over to arbitrary number fields. Here the author shows that one can get `sharp' asymptotic estimates for \(\psi_ K(cx,y)-\psi_ K(x,y)\), where \(1\leq c\leq y\), the range depending on the range in which one can give sharp estimates for the number of prime ideals whose norm lie in short intervals. The author's result is directly analogous to that of \textit{A. Hildebrand} for \(\mathbb{Q}\) [Trans. Am. Math. Soc. 297, 729-751 (1986; Zbl 0611.10028)].
scientific article; zbMATH DE number 219913
scientific article