Stephens' constant explained

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

a and b be two multiplicatively independent integers, that is, a ^{m} b ^{n} ≠ 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 ofC _{S} = \prod _{p} \left( 1 - (p)/(p ^{3-1}) \right) = 0 . 57595996889294543964316337549249669 \ldots C _{A} that arises in the study of primitive roots.^{[2] [3]} 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

Notes and References

1. Stephens . P. J. . Prime Divisor of Second-Order Linear Recurrences, I. . . 8 . 313–332 . 1976 . 3 . 10.1016/0022-314X(76)90010-X. free .
2. Pieter . Moree . Peter . Stevenhagen . A two-variable Artin conjecture . . 85 . 2000 . 2 . 291 - 304 . 10.1006/jnth.2000.2547 . math/9912250. 119739429 .
3. Pieter . Moree . Approximation of singular series and automata . . 101 . 2000 . 3 . 385 - 399 . 10.1007/s002290050222. 121036172 .