Uniform bounds for the density in Artin's conjecture on primitive roots
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911762146656256 |
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| author | Perucca, Antonella Shparlinski, Igor E. |
| author_facet | Perucca, Antonella Shparlinski, Igor E. |
| contents | We consider Artin's conjecture on primitive roots over a number field $K$, reducing an algebraic number $α\in K^\times$. Under the Generalised Riemann Hypothesis, there is a density ${\mathrm{dens}}(α)$ counting the proportion of the primes of $K$ for which $α$ is a primitive root. This density ${\mathrm{dens}}(α)$ is a rational multiple of an Artin constant $A(τ)$ that depends on the largest integer $τ\geq 1$ such that $α\in (K^\times)^τ$. The aim of this paper is bounding the ratio ${\mathrm{dens}}(α)/A(τ)$, under the assumption that ${\mathrm{dens}}(α)\neq 0$. Over $\mathbb Q$, this ratio is between $2/3$ and $2$, these bounds being optimal. For a general number field $K$ we provide upper and lower bounds that only depend on $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform bounds for the density in Artin's conjecture on primitive roots Perucca, Antonella Shparlinski, Igor E. Number Theory We consider Artin's conjecture on primitive roots over a number field $K$, reducing an algebraic number $α\in K^\times$. Under the Generalised Riemann Hypothesis, there is a density ${\mathrm{dens}}(α)$ counting the proportion of the primes of $K$ for which $α$ is a primitive root. This density ${\mathrm{dens}}(α)$ is a rational multiple of an Artin constant $A(τ)$ that depends on the largest integer $τ\geq 1$ such that $α\in (K^\times)^τ$. The aim of this paper is bounding the ratio ${\mathrm{dens}}(α)/A(τ)$, under the assumption that ${\mathrm{dens}}(α)\neq 0$. Over $\mathbb Q$, this ratio is between $2/3$ and $2$, these bounds being optimal. For a general number field $K$ we provide upper and lower bounds that only depend on $K$. |
| title | Uniform bounds for the density in Artin's conjecture on primitive roots |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.11589 |