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
$1\le  a\le  \exp\left((\log \log x)^2\right)$. This improves on a result of Stephens (1969) which applies to the
range   $1\le  a\le \exp\left( 6 ( \log x \log  \log x)^{1/2}\right )$. 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). 

Joint work with Oleksiy Klurman and Joni Teräväinen.