Artin's conjecture on average and short character sums

3:00pm, Wednesday 9th April

Abstract

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 $1\le  a\le  \exp((\log \log x)^2)$. 
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). 

Joint work with Oleksiy Klurman and Joni Teravainen