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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2208.13240 |
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| _version_ | 1866915174200377344 |
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| author | Péringuey, Paul |
| author_facet | Péringuey, Paul |
| contents | An integer is a primitive root modulo a prime $p$ if it generates the whole multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$. In 1927 Artin conjectured that an integer $a$ which is not $-1$ or a square is a primitive root for infintely many primes, and that the set of those primes has a positive asymptotic density among all primes. This conjectured was proved, under the generalized Riemann hypothesis (GRH), in 1967 by Hooley. More generally, an integer is called a generalized primitive root modulo $n$ if it generates a subgroup of $(\mathbb{Z}/n\mathbb{Z})^*$ of maximal size. Li and Pomerance showed, under GRH, that the set of integers for which a given integer is a generalized primitive root doesn't have an asymptotic density among all integers. We study here the set of the $\ell$-almost primes, i.e. integers with at most $\ell$ prime factors, for which a given integer $a\in\mathbb{Z}\backslash\{-1\}$, which is not a square, is a generalized primitive root, and we prove, under GRH, that this set has an asymptotic density among all the $\ell$-almost primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13240 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sur une généralisation de la conjecture d'Artin parmi les presque-premiers Péringuey, Paul Number Theory An integer is a primitive root modulo a prime $p$ if it generates the whole multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$. In 1927 Artin conjectured that an integer $a$ which is not $-1$ or a square is a primitive root for infintely many primes, and that the set of those primes has a positive asymptotic density among all primes. This conjectured was proved, under the generalized Riemann hypothesis (GRH), in 1967 by Hooley. More generally, an integer is called a generalized primitive root modulo $n$ if it generates a subgroup of $(\mathbb{Z}/n\mathbb{Z})^*$ of maximal size. Li and Pomerance showed, under GRH, that the set of integers for which a given integer is a generalized primitive root doesn't have an asymptotic density among all integers. We study here the set of the $\ell$-almost primes, i.e. integers with at most $\ell$ prime factors, for which a given integer $a\in\mathbb{Z}\backslash\{-1\}$, which is not a square, is a generalized primitive root, and we prove, under GRH, that this set has an asymptotic density among all the $\ell$-almost primes. |
| title | Sur une généralisation de la conjecture d'Artin parmi les presque-premiers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2208.13240 |