Primitive Root Conjecture in Arithmetic Progressions
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866914059158290432 |
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| author | Carella, N. A. |
| author_facet | Carella, N. A. |
| contents | Let $x\geq 1$ be a large number, and let $1 \leq a <q $ be integers such that $\gcd(a,q)=1$ and $q=O(\log^c)$ with $c>0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a fixed primitive root $u\ne \pm 1, v^2$ has the asymptotic formula $π_u(x,q,a)=δ(u,q,a)x/ \log x +O(x/\log^b x),$ where $δ(u,q,a)>0$ is the density, and $b=b(c)>1$ is a constant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1701_03188 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Primitive Root Conjecture in Arithmetic Progressions Carella, N. A. General Mathematics 11A07, 11N37 Let $x\geq 1$ be a large number, and let $1 \leq a <q $ be integers such that $\gcd(a,q)=1$ and $q=O(\log^c)$ with $c>0$ constant. This note proves that the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$ with a fixed primitive root $u\ne \pm 1, v^2$ has the asymptotic formula $π_u(x,q,a)=δ(u,q,a)x/ \log x +O(x/\log^b x),$ where $δ(u,q,a)>0$ is the density, and $b=b(c)>1$ is a constant. |
| title | Primitive Root Conjecture in Arithmetic Progressions |
| topic | General Mathematics 11A07, 11N37 |
| url | https://arxiv.org/abs/1701.03188 |