Sur une g\'en\'eralisation de la conjecture d'Artin parmi les presque-premiers

TL;DR

This paper extends the study of primitive roots to lmost primes, showing under GRH that the set of such lmost primes for which a fixed integer is a generalized primitive root has a positive asymptotic density.

Contribution

It proves, under GRH, that the set of lmost primes where a fixed integer is a generalized primitive root has an asymptotic density, extending previous results to this subset.

Findings

Under GRH, the set of lmost primes with a fixed generalized primitive root has positive density.

The result generalizes classical primitive root conjectures to lmost primes.

The study connects primitive root theory with the distribution of lmost primes.

Abstract

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.