Higher Order Dualities between Prime Ideals

2025/12/26 by Sroyon Sengupta, Sengupta, Sroyon
Mathematics · #11R04 #11R44 #11R45 #11R47 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.2512.22346

openalex publication_date 2025/12/26 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

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.

Citations

• Algebraic analogues of results of Alladi-Johnson using the Chebotarev Density Theorem
• Duality between prime factors and the Prime Number Theorem for Arithmetic Progressions -- II
• Formulas for Chebotarev densities of Galois extensions of number fields

Related

• Möbius formulas for densities of sets of prime ideals
• Formulas for Chebotarev densities of Galois extensions of number fields
• An effective version of Chebotarev's density theorem
• Explicit counting of ideals and a Brun-Titchmarsh inequality for the\n Chebotarev Density Theorem
• The least prime ideal in the Chebotarev Density Theorem