Stephens' constant

Stephens' constant expresses the density of certain subsets of the prime numbers.^[1][2] 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 validity of 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

${\displaystyle C_{S}=\prod _{p}\left(1-{\frac {p}{p^{3}-1}}\right)=0.57595996889294543964316337549249669\ldots }$(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.^[3][4]

${\displaystyle C_{S}=\prod _{p}\left(C_{A}+\left({{1-p^{2}} \over {p^{2}(p-1)}}\right)\right)\left({{p} \over {(p+1+{{1} \over {p}})}}\right)}$

See also

• Euler product
• Twin prime constant