Bertrand’s postulate for number fields

Thomas A. Hulse¹ ; M. Ram Murty²

¹ Department of Mathematics and Statistics Colby College Waterville, ME 04901, U.S.A.
² Department of Mathematics and Statistics Queen’s University Kingston, ON, Canada, K7L 3N6

Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 165-180.

Résumé

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 $x$.

Détail

DOI : 10.4064/cm7048-9-2016

Keywords: consider algebraic number field its ring integers mathcal there exists smallest prime ideal mathfrak mathcal norm mathfrak interval generalization bertrand postulate number fields paper produce bounds terms invariants effective prime ideal theorem due lagarias odlyzko nbsp bound obtained asymptotic estimate number ideals mathcal norm nbsp