On Artin's conjecture on average and short character sums

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-05-23 DOI:10.1112/blms.70103

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 $1 ⩽ a ⩽ \exp ({(\log \log x)}^{2})$ . 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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论马丁关于平均和短字符和的猜想

设N a (x) $N_a(x)$表示x以下质数的个数$x$其中整数a$a$是一个原始根。我们证明了na (x) $N_a(x)$对于几乎所有的1≤a都满足由Artin猜想预测的渐近≤exp ((log log x) 2) $1\leqslant a\leqslant \exp ((\log \log x)^2)$。这改进了Stephens（1969）的结果。证明中的一个关键因素是一个新的整数短字符和估计，改进了Garaev（2006）的结果的范围。

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