On Artin's conjecture on average and short character sums
Oleksiy Klurman, Igor E. Shparlinski, Joni Teräväinen
TL;DR
The paper studies Artin's conjecture on average by analyzing the primes $p\le x$ for which a fixed integer $a$ is a primitive root, via the counting function $N_a(x)$. It introduces a new short Dirichlet character-sum bound that holds for almost all moduli up to $x$, enabling a substantially stronger almost-all range for $a$: one can take $1\le a\le \exp((\log\log x)^2)$ and still have $N_a(x)$ match the conjectured main term $A(h)\pi(x)$ (with $h$ the maximal exponent such that $a$ is an $h$-th power). The method combines Stephens' framework with the short-sum estimate and an anatomy of integers to obtain a second-moment bound $\sum_{|a|\le y}|N_a(x)-A\pi(x)|^2 \ll y\pi(x)^2/(\log x)^{3D}$, yielding an almost-sure asymptotic via Chebyshev. This advances unconditional understanding of Artin's conjecture on average and showcases how short character-sum techniques can sharpen averaging results in analytic number theory.
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).