Artin's conjecture says that an integer *a* that is not -1 or a square number is a primitive root mod p for an infinite number of primes. It gives a density to the set of primes for which *a* is a primitive root.

Here is the conjecture take from wikipedia article Artin's conjecture:

Let a be an integer which is not a perfect square and not -1. 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 Schnirelmann density inside the set of primes. In particular, S(a) is infinite.

2. under the condition that a be squarefree, this density is independent of a and equals the Artin constant which can be expressed as an infinite product

I would guess that the Schnirelmann density obtain is greater when a particular *a* has a large set of primes for which *a* is a primitive root.