On Artin's conjecture on average and short character sums

TL;DR

This paper proves that for most integers within a certain range, the distribution of primes for which these integers are primitive roots aligns with Artin's conjecture, using new short character sum estimates.

Contribution

It establishes the asymptotic behavior of primitive root counts for almost all integers up to a specified bound, improving previous results with novel character sum bounds.

Findings

Proves Artin's conjecture holds on average for most integers in a specific range.

Introduces a new short character sum estimate over integers.

Enhances the range of applicability compared to previous results.

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).