Artin's conjecture on primitive roots

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This page discusses a conjecture of Emil Artin on primitive roots. For the conjecture of Artin on L-functions, see Artin L-function.

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

S(a) has a positive density inside the set of primes. In particular, S(a) is infinite.
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

$C_{Artin}=\prod_{p\ \mathrm{prime},\ p>0} \left(1-\frac{1}{p(p-1)}\right) = 0.3739558136\ldots$

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 (sequence A001122 in OEIS)

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.^{[citation needed]} In 1967, Hooley published a conditional proof for the conjecture, assuming certain cases of the Generalized Riemann hypothesis.^[1] In 1984, R. Gupta and M. Ram Murty showed unconditionally that Artin's conjecture is true for infinitely many a using sieve methods.^[2] 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.^[3] 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.

[edit] See also

Brown-Zassenhaus conjecture
Cyclic number

[edit] References

^ Hooley, Christopher (1967). "On Artin's conjecture." J. Reine Angew. Math. 225, 209-220.
^ Gupta, Rajiv & Murty, M. Ram (1984). "A remark on Artin's conjecture." Invent. Math. 78 (1), 127-130.
^ Heath-Brown, D.R. (1986). "Artin's conjecture for primitive roots." Quart. J. Math. Oxford Ser. (2) 37, 27-38.