Refinements of Artin's primitive root conjecture
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910847494782976 |
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| author | Goldmakher, Leo Martin, Greg Péringuey, Paul |
| author_facet | Goldmakher, Leo Martin, Greg Péringuey, Paul |
| contents | A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit formula for the proportion; this formula is well-supported by computations and is known to hold on a generalized Riemann hypothesis, but remains open. In this paper we propose several conjectures that capture the finer properties of the distribution of the order of $a$ (mod $p$) as $p$ varies over primes; these assertions contain Artin's original conjecture as a special case. We prove these conjectures assuming the generalized Riemann hypothesis, as well as weaker versions unconditionally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refinements of Artin's primitive root conjecture Goldmakher, Leo Martin, Greg Péringuey, Paul Number Theory 11A07, 11N05, 11A25 (Primary) 11R20, 11Y60 (Secondary) A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit formula for the proportion; this formula is well-supported by computations and is known to hold on a generalized Riemann hypothesis, but remains open. In this paper we propose several conjectures that capture the finer properties of the distribution of the order of $a$ (mod $p$) as $p$ varies over primes; these assertions contain Artin's original conjecture as a special case. We prove these conjectures assuming the generalized Riemann hypothesis, as well as weaker versions unconditionally. |
| title | Refinements of Artin's primitive root conjecture |
| topic | Number Theory 11A07, 11N05, 11A25 (Primary) 11R20, 11Y60 (Secondary) |
| url | https://arxiv.org/abs/2502.19601 |