Stephens' constant

In number theory, Stephens' constant expresses the density of certain subsets of the prime numbers.¹² Let ${\displaystyle a}$ and ${\displaystyle b}$ be two multiplicatively independent integers, that is, ${\displaystyle a^{m}b^{n}\neq 1}$ except when both ${\displaystyle m}$ and ${\displaystyle n}$ equal zero. Consider the set ${\displaystyle T(a,b)}$ of prime numbers ${\displaystyle p}$ such that ${\displaystyle p}$ evenly divides ${\displaystyle a^{k}-b}$ for some power ${\displaystyle k}$. Assuming the generalized Riemann hypothesis, the density of the set ${\displaystyle T(a,b)}$ relative to the set of all primes is a rational multiple of

${\displaystyle C_{S}=\prod _{p}{\bigg (}1-{\frac {p}{p^{3}-1}}{\bigg )}=0.57595996889294543964316337549249669\ldots }$(sequence A065478 in the OEIS)

Stephens' constant is closely related to the Artin constant ${\displaystyle C_{A}}$ that arises in the study of primitive roots:³⁴

${\displaystyle C_{S}=\prod _{p}{\bigg (}C_{A}+{\frac {1-p^{2}}{p^{2}(p-1)}}{\bigg )}{\bigg (}{\frac {p}{p+1+1/p}}{\bigg )}.}$