A generalised Kummer's Conjecture

TL;DR

This paper extends Kummer's Conjecture to all cyclotomic fields and explores its validity under the Elliott-Halberstam Conjecture, showing it holds for almost all n but fails infinitely often.

Contribution

It generalizes Kummer's Conjecture to arbitrary conductors and links its truth to the Elliott-Halberstam Conjecture, providing new insights into class number growth.

Findings

Conjecture holds for almost all n under Elliott-Halberstam

Conjecture is false for infinitely many n

Connections established between class numbers and prime distribution

Abstract

Kummer's Conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's Conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott-Halberstam Conjecture implies that this Generalised Kummer's Conjecture is true for almost all n but is false for infinitely many n.