Higher Order Dualities between Prime Ideals

TL;DR

This work extends Alladi-type dualities to prime ideals in number fields, establishing a general higher-order duality (beginning with a second-order case) and using it to derive new Chebotarev-density-type formulas in arbitrary finite Galois extensions. The methodology combines a second-order duality with the strong form of the Chebotarev Density Theorem to obtain quantitative density statements, along with explicit bounds for sums involving the generalized Mobius function $\mu_K(I)$ and the prime-ideal count $\omega_K(I)$. It also develops robust counting tools for ideals, such as $\Psi_K(X,Y)$ and $\Psi_{K,2}(X,Y)$, and extends the duality to arbitrary fixed sets of prime ideals, providing a versatile framework for density-type results across number fields. Overall, the paper unifies and generalizes prior integer- and algebraic-case dualities (Alladi, Dawsey, Sweeting–Woo, Johnson, Se25) and equips researchers with a broad, quantitative toolkit for duality-based density identities in algebraic number theory.

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 Möbius 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.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Theorem 2.4