In number theory, Stephens' constant expresses the density of certain subsets of the prime numbers. Let a {\displaystyle a} and b {\displaystyle b} be two multiplicatively independent integers, that is, a m b n ≠ 1 {\displaystyle a^{m}b^{n}\neq 1} except when both m {\displaystyle m} and n {\displaystyle n} equal zero. Consider the set T ( a , b ) {\displaystyle T(a,b)} of prime numbers p {\displaystyle p} such that p {\displaystyle p} evenly divides a k − b {\displaystyle a^{k}-b} for some power k {\displaystyle k} . Assuming the generalized Riemann hypothesis, the density of the set T ( a , b ) {\displaystyle T(a,b)} relative to the set of all primes is a rational multiple of
C S = ∏ p ( 1 − p p 3 − 1 ) = 0.57595996889294543964316337549249669 … {\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 C A {\displaystyle C_{A}} that arises in the study of primitive roots:
C S = ∏ p ( C A + 1 − p 2 p 2 ( p − 1 ) ) ( p p + 1 + 1 / p ) . {\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 )}.}
See also Euler product Twin prime constant
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