论文标题
Sur unegénémaritationd de la to d'Artin parmi les presque-Premiers
Sur une généralisation de la conjecture d'Artin parmi les presque-premiers
论文作者
论文摘要
整数是一个原始的root modulo，如果生成整个乘法组$（\ Mathbb {z}/p \ Mathbb {z}）^*$，则它是Prime $ P $。 1927年，Artin猜想，不是$ -1 $或正方形的整数$ a $是无所不能的许多素数的原始根，并且这些素数的集合在所有素数中具有正渐近密度。 Hooley于1967年根据普遍的Riemann假设（GRH）证明了这一猜想。更一般而言，如果整数生成$（\ Mathbb {z}/n \ Mathbb {z}）^*$的最大大小的$（\ Mathbb {z}/n \ Mathbb {z}/n \ Mathbb {z}/n \ Mathbb {z}/n \ mathbb { Li和Pomerance在GRH下表明，给定整数是广义原始根的整数集合并不具有所有整数之间的渐近密度。我们在这里研究一组$ \ ell $ - 最多具有$ \ ell $ prime因素的整数，给定的整数$ a \ in \ mathbb {z} \ backslash \ backslash \ \ { - 1 \} $，不是一个正方形的，这是一个普遍的原始根，我们都在grh grh，grh of grh and of there and。 $ \ ell $ - 几乎是素数。
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.
