Stephens' constant

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

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

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

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

See also

• Euler product
• Twin prime constant