Bertrand’s postulate for number fields
Volume 147 / 2017
Colloquium Mathematicum 147 (2017), 165-180
MSC: Primary 11R44; Secondary 11R42.
DOI: 10.4064/cm7048-9-2016
Published online: 14 December 2016
Abstract
Consider an algebraic number field, $K$, and its ring of integers, $\mathcal {O}_K$. There exists a smallest $B_K \gt 1$ such that for any $x \gt 1$ we can find a prime ideal, $\mathfrak {p}$, in $\mathcal {O}_K$ with norm $N(\mathfrak {p})$ in the interval $[x,B_Kx]$. This is a generalization of Bertrand’s postulate to number fields, and in this paper we produce bounds on $B_K$ in terms of the invariants of $K$ from an effective prime ideal theorem due to Lagarias and Odlyzko (1977). We also show that a bound on $B_K$ can be obtained from an asymptotic estimate for the number of ideals in $\mathcal {O}_K$ with norm less than $x$.
