Existence of a limiting density for primitive roots among polynomial primes

From papers

Let f(X)Z[X]f(X)\in\mathbb Z[X] represent infinitely many primes, and let gGg\in G, where GG is the set of integers under consideration as possible primitive roots. Define

Rg(f;x)=#{px:f(m)=p for some m and g is a primitive root modulo p}#{px:f(m)=p for some m}.R_g(f;x)=\frac{\#\{p\le x:f(m)=p\text{ for some }m\text{ and }g\text{ is a primitive root modulo }p\}}{\#\{p\le x:f(m)=p\text{ for some }m\}}.

Primitive-root density conjecture. The quotient Rg(f;x)R_g(f;x) tends to a limit as xx tends to infinity; denote this conjectural density by δg(f)\delta_g(f). This predicts that the relative proportion of polynomial prime values for which gg is a primitive root exists. The paper presents it as numerical evidence and a heuristic expectation; no proof or resolution is supplied.

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Sources & referencesView supporting material

Primary source

Pieter Moree, “Primitive root producing quadratics”, arXiv:math/0406033 (2004).

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