There are at least two statements which go by the name of Artin's conjecture. The first is the Riemann Hypothesis. The second states that every Integer not equal to or a Square Number is a primitive root modulo for infinitely many and proposes a density for the set of such which are always rational multiples of a constant known as Artin's Constant. There is an analogous theorem for functions instead of numbers which has been proved by Billharz (Shanks 1993, p. 147).
See also Artin's Constant, Riemann Hypothesis
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 31, 80-83, and 147, 1993.