Bertrand's Postulate for Number Fields

TL;DR

This paper generalizes Bertrand's postulate to algebraic number fields, establishing bounds on the minimal constant ensuring prime ideals with norms in specific intervals, based on invariants of the field and effective prime ideal theorems.

Contribution

It provides explicit bounds on the constant $B_K$ in number fields using invariants and prime ideal theorems, extending classical results to algebraic number theory.

Findings

Bounds on $B_K$ in terms of field invariants

Effective prime ideal theorem application

Relation between $B_K$ and ideal counting asymptotics

Abstract

Consider an algebraic number field, $K$, and its ring of integers, $\mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ 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. We also show that a bound on $B_K$ can be obtained from an asymptotic estimate for the number of ideals in $\mathcal{O}_K$ less than $x$.