Higher Order Dualities between Prime Ideals

TL;DR

This paper explores higher order dualities between prime ideals in number rings, deriving new formulas for Chebotarev Density and providing estimates for related sums using duality identities.

Contribution

It extends previous duality theories to higher orders and applies them to derive new formulas and estimates for prime ideal distributions in number rings.

Findings

Derived a new formula for Chebotarev Density involving generalized M"obius function

Provided estimates for sums of prime ideal counting functions

Extended duality concepts to a more general setting

Abstract

Extending the works of Alladi and Sweeting and Woo, we state and prove the general higher order duality between prime ideals in number rings. We then use the second order duality to obtain the a new formula for the Chebotarev Density involving sums of the generalized M\"obius function and the prime ideal counting function. We also provide two estimates of such sums as an application of the duality identity. A discussion of the duality in a slightly more general setting is done at the end.