Bertrand's Postulate for Number Fields

2015/08/04 by Hulse, Thomas A., Murty, M. Ram
#11R42 (secondary) #11R44 (primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1508.00887

Abstract

Consider an algebraic number field, K, and its ring of integers, O_K. There exists a smallest B_K>1 such that for any x>1 we can find a prime ideal, \mathfrakp, in O_K with norm N(\mathfrakp) 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 O_K less than x.

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