On Artin's conjecture on average and short character sums

Let N_a(x) denote the number of primes up to x for which the integer a 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).