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Artin's conjecture on primitive roots

In mathematics, the Artin conjecture is a conjecture on the set of primes p modulo which a given integer a > 1 is a primitive root. The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary.

The precise statement is as follows. 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

$C_\left\{Artin\right\}=prod_\left\{p mathrm\left\{prime\right\}, p>0\right\} left\left(1-frac\left\{1\right\}\left\{p\left(p-1\right)\right\}right\right) = 0.3739558136ldots$
Similar product formulas exist for the density when a contains a square factor.

For example, take a = 2. The conjecture claims that the set of primes p for which 2 is a primitive root has the above density C. The set of such primes is

S(2)={3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, ...}
It has 38 elements smaller than 500 and there are 95 primes smaller than 500. The ratio (which conjecturally tends to C) is 38/95=0.41051...

To prove the conjecture, it is sufficient to do so for prime numbers a. In 1967, Hooley published a conditional proof for the conjecture, assuming certain cases of the Generalized Riemann hypothesis. In 1984, R. Gupta and M. Ram Murty showed unconditionally that Artin's conjecture is true for infinitely many a using sieve methods. Roger Heath-Brown improved on their result and showed unconditionally that there are at most two exceptional prime numbers a for which Artin's conjecture fails. This result is not constructive, as far as the exceptions go. For example, it follows from the theorem of Heath-Brown that one out of 3, 5, and 7 is a primitive root modulo p for infinitely many p. But the proof does not provide us with a way of computing which one. In fact, there is not a single value of a for which the Artin conjecture is known to hold.