Small Prime Gaps in Abelian Number Fields

TL;DR

This paper extends the study of small prime gaps to primes splitting completely in abelian number fields, providing both conditional and unconditional results, including a new proof for quadratic extensions with class number 1.

Contribution

It introduces new results on small prime gaps in abelian number fields, generalizing previous work and offering a novel proof for quadratic fields with class number one.

Findings

Conditional small prime gap results assuming Elliott-Halberstam conjecture

Unconditional small prime gap results in abelian fields

A new proof for quadratic extensions with class number 1

Abstract

We prove an analogue of a result by Goldston, Pintz and Yildirim for small gaps between primes that split completely in an abelian number field. We prove both a conditional result assuming the Elliott-Halberstam conjecture, and an unconditional result. We also give another proof of the same result in the special case of a quadratic extension of class number 1, which relies on a generalization of the Bombieri-Vinogradov theorem for quadratic number fields.