Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Goldmakher, Leo, Martin, Greg, Péringuey, Paul
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416149100
Cites:
Cited by:
Related to: