Artin's Conjecture On Primitive Roots - Formulation

Formulation

Let a be an integer which is not a perfect square and not −1. Write a = a0b2 with a0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then

  1. S(a) has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.
  2. Under the conditions that a is not a perfect power and that a0 is not congruent to 1 modulo 4, this density is independent of a and equals Artin's constant which can be expressed as an infinite product
    (sequence A005596 in OEIS).

Similar conjectural product formulas exist for the density when a does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of CArtin.

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