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

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 .