Stephens' constant

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

[math]\displaystyle{ C_S = \prod_p \left(1 - \frac{p}{p^3-1} \right) = 0.57595996889294543964316337549249669\ldots }[/math](sequence A065478 in the OEIS)

Stephens' constant is closely related to the Artin constant [math]\displaystyle{ C_A }[/math] that arises in the study of primitive roots.

[math]\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) }[/math]

See also

• Euler product
• Twin prime constant