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

References

1. Stephens, P. J. (1976). "Prime Divisor of Second-Order Linear Recurrences, I." Journal of Number Theory. 8 (3): 313–332. doi:10.1016/0022-314X(76)90010-X.
2. Weisstein, Eric W. "Stephens' Constant". MathWorld.
3. Moree, Pieter; Stevenhagen, Peter (2000). "A two-variable Artin conjecture". Journal of Number Theory. 85 (2): 291–304. arXiv:math/9912250. doi:10.1006/jnth.2000.2547. S2CID 119739429.
4. Moree, Pieter (2000). "Approximation of singular series and automata". Manuscripta Mathematica. 101 (3): 385–399. doi:10.1007/s002290050222. S2CID 121036172.