Uniformly counting primes with a given primitive root and in an arithmetic progression

Michel Zoeteman

Type: Article

Publication Date: 2019-05-27

Citations: 0

DOI: https://doi.org/10.1142/s1793042119501161

Abstract

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin’s primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.

Locations

International Journal of Number Theory - View
arXiv (Cornell University) - View - PDF

Similar Works

Action	Title	Year	Authors
+	Uniformly counting primes with a given primitive root and in an arithmetic progression	2018	Michel Zoeteman
+	Uniformly counting primes with a given primitive root and in an arithmetic progression	2018	Michel Zoeteman
+	Uniform bounds for the density in Artin's conjecture on primitive roots	2024	Antonella Perucca Igor E. Shparlinski
+	Uniform bounds for the density in Artin's conjecture on primitive roots	2025	Antonella Perucca Igor E. Shparlinski
+	On the density of primes in arithmetic progression having a prescribed primitive root	1999	Pieter Moree
+ PDF Chat	The correction factor in Artin's primitive root conjecture	2003	Peter Stevenhagen
+	Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis	2021	Anne-Maria Ernvall-Hytönen Neea Palojärvi
+	Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis	2020	Anne-Maria Ernvall-Hytönen Neea Palojärvi
+	Artin's conjecture on primes with prescribed primitive roots	1977	Jr. Lenstra
+	Artin's conjecture on primes with prescribed primitive roots	1978	H. W. Lenstra
+	A problem analogous to Artin's conjecture for primitive roots and its applications	1991	Leo Murata
+ PDF Chat	On primes in arithmetic progression having a prescribed primitive root.II	2008	Pieter Moree
+	Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture	2019	Su Hu Min-Soo Kim Pieter Moree Min Sha
+ PDF Chat	ON A CONJECTURE OF ERDŐS	2012	Adam Tyler Felix M. Ram Murty
+	Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture	2018	Su Hu Min-Soo Kim Pieter Moree Min Sha
+	Divisor problem in arithmetic progressions modulo a prime power	2016	Kui Liu Igor E. Shparlinski Tianping Zhang
+	On Burgess' Bound for Primitive Roots Modulo Primes and an Application to Γ(p)	1981	Emil Grosswald
+ PDF Chat	Artin's primitive root conjecture for quadratic fields	2002	Hans Roskam
+	The Artin–Carmichael Primitive Root Problem on Average	2009	Shuguang Li Carl Pomerance
+	On the Average of the Least Primitive Root Modulo $p$(Analytic Number Theory)	1996	Leo Murata

Works That Cite This (0)

Action	Title	Year	Authors

Works Cited by This (8)

Action	Title	Year	Authors
+	On Artin's conjecture.	1967	C. Hooley
+ PDF Chat	The correction factor in Artin's primitive root conjecture	2003	Peter Stevenhagen
+ PDF Chat	On Artin's conjecture and Euclid's algorithm in global fields	1977	H. W. Lenstra
+ PDF Chat	Quelques applications du théorème de densité de Chebotarev	1981	Jean-Pierre Serre
+ PDF Chat	On primes in arithmetic progression having a prescribed primitive root.II	2008	Pieter Moree
+	Zur additiven Zahlentheorie. II.	1936	Arnold Walfisz
+	Graduate Texts in Mathematics	2020	J. H. van Lint
+ PDF Chat	Character sums for primitive root densities	2014	H. W. Lenstra Peter Stevenhagen Pieter Moree