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Bertrand’s postulate for number fields

Volume 147 / 2017

Thomas A. Hulse, M. Ram Murty 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 $x$.

Authors

  • Thomas A. HulseDepartment of Mathematics and Statistics
    Colby College
    Waterville, ME 04901, U.S.A.
    e-mail
  • M. Ram MurtyDepartment of Mathematics and Statistics
    Queen’s University
    Kingston, ON, Canada, K7L 3N6
    e-mail

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