On Artin's conjecture on average and short character sums

Abstract

<p>Let N_a(x) denote the number of primes up to <i>x</i> for which the integer <i>a</i> is a primitive root. We show that N_a(x) satisfies the asymptotic predicted by Artin's conjecture for almost all 1 ⩽ a ⩽ exp((log log x)²). This improves on a result of Stephens (1969). A key ingredient in the proof is a new short character sum estimate over the integers, improving on the range of a result of Garaev (2006).<br></p>