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arXiv · 2412.13355

On Artin's conjecture on average and short character sums

Oleksiy Klurman·Igor E. Shparlinski·Joni Teräväinen
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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).

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Oleksiy Klurman, Igor E. Shparlinski, Joni Teräväinen. 2024-12-17. On Artin's conjecture on average and short character sums. https://doi.org/10.1112/blms.70103

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