Work

Uniform bounds for the density in Artin's conjecture on primitive roots

Year: 2025

Type: article

Abstract: We consider Artin's conjecture on primitive roots over a number field , reducing an algebraic number . Under the Generalised Riemann Hypothesis, there is a density counting the proportion of the primes of for which is a primitive root. This density is a rational multiple of an Artin constant that depends on the largest integer such that . The aim of this paper is bounding the ratio , under the assumption that . Over , this ratio is between and 2, these bounds being optimal. For a general number field , we provide upper and lower bounds that only depend on .

Source: Bulletin of the London Mathematical Society

DOI: 10.1112/blms.70011

Authors: Antonella Perucca, Igor E. Shparlinski

Institutions: University of Luxembourg, UNSW Sydney

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,930

Funder: Australian Research Council