Unconditional results for Artin-type problems over number fields
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915442281414656 |
|---|---|
| author | Sgobba, Pietro |
| author_facet | Sgobba, Pietro |
| contents | Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unconditional results for Artin-type problems over number fields Sgobba, Pietro Number Theory 11R45 (Primary) 11R44 (Secondary) Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. 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). |
| title | Unconditional results for Artin-type problems over number fields |
| topic | Number Theory 11R45 (Primary) 11R44 (Secondary) |
| url | https://arxiv.org/abs/2508.08996 |