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arXiv · 2502.19601

Refinements of Artin's primitive root conjecture

Leo Goldmakher·Greg Martin·Paul Péringuey
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Abstract

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.

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Leo Goldmakher, Greg Martin, Paul Péringuey. 2025-02-26. Refinements of Artin's primitive root conjecture. https://arxiv.org/abs/2502.19601

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