Artin's conjecture on primitive roots

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In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a perfect square nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.

The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of 2021. In fact, there is no single value of a for which Artin's conjecture is proved.

Formulation

Let a be an integer that 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 the conjecture states

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 (sequence A085397 in the OEIS), this density is independent of a and equals Artin's constant, which can be expressed as an infinite product
${\displaystyle C_{\mathrm {Artin} }=\prod _{p\ \mathrm {prime} }\left(1-{\frac {1}{p(p-1)}}\right)=0.3739558136\ldots }$ (sequence A005596 in the OEIS).

Similar conjectural product formulas [1] 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.

Example

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 CArtin. The set of such primes is (sequence A001122 in the 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 CArtin) is 38/95 = 2/5 = 0.4.

Partial results

In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized Riemann hypothesis.[2]

Without the generalized Riemann hypothesis, there is no single value of a for which Artin's conjecture is proved. D. R. Heath-Brown proved (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes p.[3] He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.

References

1. ^ Michon, Gerard P. (2006-06-15). "Artin's Constant". Numericana.
2. ^ Hooley, Christopher (1967). "On Artin's conjecture". J. Reine Angew. Math. 225: 209–220. doi:10.1515/crll.1967.225.209. MR 0207630.
3. ^ D. R. Heath-Brown (March 1986). "Artin's Conjecture for Primitive Roots". The Quarterly Journal of Mathematics. 37 (1): 27–38. doi:10.1093/qmath/37.1.27.