Primitive roots for Pjateckii-Šapiro primes

Jyothsnaa Sivaraman

doi:10.5802/jtnb.1152

Jyothsnaa Sivaraman ¹

¹ University of Toronto, Toronto Ontario, Canada - M5T 3J1.

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 83-94.

Abstract
Résumé

For any non-integral positive real number $c$ , any sequence ${(⌊ n^{c} ⌋)}_{n}$ is called a Pjateckii-Šapiro sequence. Given a real number $c$ in the interval $(1, \frac{12}{11})$ , it is known that the number of primes in this sequence up to $x$ has an asymptotic formula. We would like to use the techniques of Gupta and Murty to study Artin’s problems for such primes. We will prove that even though the set of Pjateckii-Šapiro primes is of density zero for a fixed $c$ , one can show that there exist natural numbers which are primitive roots for infinitely many Pjateckii-Šapiro primes for any fixed $c$ in the interval $(1, \frac{\sqrt{77}}{7} - \frac{1}{4})$ .

Pour tout nombre réel positif non entier $c$ , la suite ${(⌊ n^{c} ⌋)}_{n}$ est appelée suite de Pjateckii-Šapiro. Étant donné un nombre réel $c$ dans l’intervalle $(1, \frac{11}{12})$ , on a une formule asymptotique pour le nombre de nombres premiers de cette suite qui sont au plus égaux à $x$ . Nous utilisons la méthode de Gupta et Murty pour étudier le problème d’Artin pour ces nombres premiers. Nous démontrons que, bien que l’ensemble de ces nombres premiers a une densité relative nulle pour $c$ donné, il existe des entiers positifs qui sont des racines primitives pour une infinité de nombres premiers de Pjateckii-Šapiro pour tout $c$ fixé dans l’intervalle $(1, \frac{\sqrt{77}}{7} - \frac{1}{4})$ .

Received: 2020-01-26
Revised: 2021-01-16
Accepted: 2021-02-01
Published online: 2021-05-21

DOI: 10.5802/jtnb.1152

Classification: 11A07, 11N05, 11N35, 11N36
Keywords: Primitive roots, Pjateckii-Šapiro sequence, primes, sieve methods

Author's affiliations:

Jyothsnaa Sivaraman ¹

¹ University of Toronto, Toronto Ontario, Canada - M5T 3J1.

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Primitive roots for {Pjateckii-\v{S}apiro} primes},
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Jyothsnaa Sivaraman. Primitive roots for Pjateckii-Šapiro primes. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 83-94. doi : 10.5802/jtnb.1152. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1152/

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