Stephens' constant

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Stephens' constant expresses the density of certain subsets of the prime numbers.[1][2] Let and be two multiplicatively independent integers, that is, except when both and equal zero. Consider the set of prime numbers such that evenly divides for some power . Assuming the validity of the generalized Riemann hypothesis, the density of the set relative to the set of all primes is a rational multiple of

(sequence A065478 in the OEIS)

Stephens' constant is closely related to the Artin constant that arises in the study of primitive roots.[3][4]

See also

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References

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  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.