Counting primes with a given primitive root, uniformly
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arXiv
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2025
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| _version_ | 1866909785215991808 |
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| author | Fan, Steve Pollack, Paul |
| author_facet | Fan, Steve Pollack, Paul |
| contents | The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $-1$ nor a perfect square, there is an explicit constant $A(g)>0$ such that the number $Π(x;g)$ of primes $p\le x$ for which $g$ is a primitive root is asymptotically $A(g)π(x)$ as $x\to\infty$, where $π(x)$ counts the number of primes not exceeding $x$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $\mathbb{Q}$. In this paper, we use GRH to establish a uniform version of the Artin--Hooley asymptotic formula. Specifically, we prove that $Π(x;g) \sim A(g) x/\log{x}$ whenever $\log{x}/\log\log{2|g|} \to \infty$, i.e., whenever $x$ tends to infinity faster than any power of $\log{(2|g|)}$. Under GRH, we also show that the least prime $p_g$ possessing $g$ as a primitive root satisfies the upper bound $p_g=O(\log^{19}(2|g|))$ uniformly for all non-square $g\ne-1$. We conclude with an application to the average value of $p_g$ and a discussion of an analogue concerning the least "almost-primitive'' root. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting primes with a given primitive root, uniformly Fan, Steve Pollack, Paul Number Theory Primary 11N05, Secondary 11A07, 11N36 The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $-1$ nor a perfect square, there is an explicit constant $A(g)>0$ such that the number $Π(x;g)$ of primes $p\le x$ for which $g$ is a primitive root is asymptotically $A(g)π(x)$ as $x\to\infty$, where $π(x)$ counts the number of primes not exceeding $x$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $\mathbb{Q}$. In this paper, we use GRH to establish a uniform version of the Artin--Hooley asymptotic formula. Specifically, we prove that $Π(x;g) \sim A(g) x/\log{x}$ whenever $\log{x}/\log\log{2|g|} \to \infty$, i.e., whenever $x$ tends to infinity faster than any power of $\log{(2|g|)}$. Under GRH, we also show that the least prime $p_g$ possessing $g$ as a primitive root satisfies the upper bound $p_g=O(\log^{19}(2|g|))$ uniformly for all non-square $g\ne-1$. We conclude with an application to the average value of $p_g$ and a discussion of an analogue concerning the least "almost-primitive'' root. |
| title | Counting primes with a given primitive root, uniformly |
| topic | Number Theory Primary 11N05, Secondary 11A07, 11N36 |
| url | https://arxiv.org/abs/2505.05601 |