2025/08/12 by Pietro Sgobba, Sgobba, Pietro
Mathematics · #11R45 (Primary) 11R44 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Benford’s Law and Fraud Detection #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2508.08996
openalex publication_date 2025/08/12 · 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^×. For all but finitely many primes \mathfrak p of K, the reduction (G \bmod \mathfrak p) generates a well-defined subgroup of the multiplicative group of the residue field at \mathfrak p, and we may consider its index. We study the primes of K for which this index lies in a given set of positive integers S. In particular, we prove that under certain convergence conditions on series associated to S this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provide asymptotic formulas for the corresponding prime-counting functions. Problems of this type are related to Artin's primitive root conjecture, which has been proven under the assumption of GRH (Hooley, 1967).