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Sur une généralisation de la conjecture d'Artin parmi les presque-premiers

Paul Péringuey

TL;DR

This paper extends Artin-type results to the setting of generalized primitive roots modulo a fixed non-square, non-$-1$ integer $a$ across $\ell$-almost primes. Under the Generalized Riemann Hypothesis, it proves that the count $\mathcal{N}_{a,\ell}(x)$ of $\ell$-almost primes $\le x$ for which $a$ is a generalized primitive root has an explicit asymptotic with main term $\dfrac{x(\log\log x)^{\ell-1}}{(\ell-1)!\,\log x}$ multiplied by a product of local factors $\big(1-W_\ell(p)\big)ig(1+V_\ell(a_1)\big)$, plus a vanishing relative error. The analysis blends a refined Hooley-type decomposition, a Selberg–Delange treatment of prime ideals in number fields, and a detailed combinatorial-sieve framework to separate conditions across multiple primes; it also provides an unconditional upper bound mirroring the main structure. The work connects generalized Artin phenomena with the Chebotarev densities of several number fields and yields an explicit density result for almost primes, highlighting the role of the algebraic data $h$, $a_1$, and the extensions $K_i$. This advances understanding of how density phenomena for primitive roots interact with the multiplicative structure of almost primes and demonstrates the power of Selberg–Delange in multi-field arithmetic problems.

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.

Sur une généralisation de la conjecture d'Artin parmi les presque-premiers

TL;DR

This paper extends Artin-type results to the setting of generalized primitive roots modulo a fixed non-square, non- integer across -almost primes. Under the Generalized Riemann Hypothesis, it proves that the count of -almost primes for which is a generalized primitive root has an explicit asymptotic with main term multiplied by a product of local factors , plus a vanishing relative error. The analysis blends a refined Hooley-type decomposition, a Selberg–Delange treatment of prime ideals in number fields, and a detailed combinatorial-sieve framework to separate conditions across multiple primes; it also provides an unconditional upper bound mirroring the main structure. The work connects generalized Artin phenomena with the Chebotarev densities of several number fields and yields an explicit density result for almost primes, highlighting the role of the algebraic data , , and the extensions . This advances understanding of how density phenomena for primitive roots interact with the multiplicative structure of almost primes and demonstrates the power of Selberg–Delange in multi-field arithmetic problems.

Abstract

An integer is a primitive root modulo a prime if it generates the whole multiplicative group . In 1927 Artin conjectured that an integer which is not 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 if it generates a subgroup of 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 -almost primes, i.e. integers with at most prime factors, for which a given integer , 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 -almost primes.
Paper Structure (22 sections, 20 theorems, 221 equations)

This paper contains 22 sections, 20 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Caractérisation des racines primitives généralisées.
  3. Approche heuristique expliquant le terme $W_\ell(p)$ des Théorèmes \ref{['thm_principal']} et \ref{['thm_inconditionnel']}
  4. Un premier découpage : le découpage initial de Hooley
  5. Une deuxième série de découpages et distinction de la majoration nécessitant l'Hypothèse de Riemann Généralisée
  6. Majoration de ${\mathcal{M}_a}(x,C_2,C_3)$ et ${\mathcal{M}_a}(x,C_3,x-1)$
  7. Majoration de ${\mathcal{M}_a}(x,C_1,C_2)$ sous GRH.
  8. Nouvelle équation fondamentale
  9. Expression de ${\mathcal{P}_a}(x,k)$
  10. Correspondance entre les $\ell$-presque premiers recherchés et les idéaux premiers de certains corps de nombres
  11. Méthode de Selberg-Delange
  12. Contrôle des termes d'erreurs
  13. Calcul du terme principal
  14. $\mathcal{P}_{a,0}(k)$ pour $k$ impair
  15. Découpage selon le diviseur pair de $k$ parmi les $k_L$.
  16. ...and 7 more sections

Key Result

Proposition 1

En reprenant les définitions de $C_2$ et $C_3$ de def_bornes_Ci, on a :

Theorems & Definitions (45)

  • proof
  • proof
  • Proposition 1
  • proof
  • Proposition 2: GRH
  • proof
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 35 more