2021/10/17 by Pietro Sgobba, Sgobba, Pietro · 1 citation
Mathematics · #11R20 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary: 11R45 #Secondary: 11R44
paper · pdf · doi:10.48550/arxiv.2110.08911
openalex publication_date 2021/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a number field, and let G be a finitely generated subgroup of K^×. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes \mathfrak p of K such that the order of (G\bmod\mathfrak p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes \mathfrak p for which the order is k-free, and those for which the order has a prescribed ℓ-adic valuation for finitely many primes ℓ. An additional condition on the Frobenius conjugacy class of \mathfrak p may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.