Artin's conjecture on primitive roots

The question

artin_primitive_roots.parts.i. Artin's Conjecture on Primitive Roots, first half. Let $a$ be an integer that is not a square number and not $-1$. Then the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ has a positive asymptotic density inside the set of primes. In particular, $S(a)$ is infinite.

artin_primitive_roots.parts.ii. Artin's Conjecture on Primitive Roots, second half. Write $a=a_0{b}^2$ where $a_0$ is squarefree. Under the conditions that $a$ is not a perfect power and $a_0\not\equiv 1(\mathrm{mod}4)$ (sequence A85397 in the OEIS), the density of the set $S(a)$ of primes $p$ such that $a$ is a primitive root modulo $p$ is independent of $a$ and equals Artin's constant.

Artin's conjecture predicts, given an integer $a$, densities of primes $p$ for which $a$ is a primitive root modulo $p$. Under certain conditions (when $a$ is not a power and its squarefree part is not $1(\mathrm{mod}4)$) the density is given by Artin's constant $$\prod _{p\text{prime}}\left(1-\frac{1}{p(p-1)}\right).$$ For more general values of $a$, this constant must be corrected by certain factors.

• When $a=b^m$, $m$ is a maximal odd power, the squarefree part of $b$ satisfies $b_0\not\equiv 1(\mathrm{mod}4)$. Then Artin's constant should be multiplied by $$\prod _{p\mid m}\frac{p(p-2)}{p^2-p-1}.$$
• When $a=b^m$, $m$ is a maximal power, the squarefree part of $b$ satisfies $b_0\equiv 1(\mathrm{mod}4)$. Then Artin's constant should be multiplied by the factor in the above bullet, as well as an additional entanglement factor from the primes dividing $\gcd (b_0,m)$ and primes dividing $b_0$: $$1-\prod _{p\mid \gcd (b_0,m)}\frac{1}{2-p}\prod _{p\mid b_0,p\nmid m}\frac{1}{1+p-p^2}.$$
• When $a=-1$ or $a$ is a square, then the density is $0$.

Note that Artin's conjecture has been proved by Hooley [Ho67] subject to the Riemann hypothesis for the Dedekind zeta functions of the Kummer fields $\mathbb{Q}({\zeta }_k,a^{1/k})$, $k$ squarefree. These fields are in general non-abelian over $\mathbb{Q}$, so this hypothesis is not covered by the Generalized Riemann Hypothesis for Dirichlet $L$-functions.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.ArtinPrimitiveRootsConjecture (2 statements).

theorem artin_primitive_roots.parts.i (a : ℤ) (ha : ¬IsSquare a) (ha' : a ≠ -1) :
    ∃ x > 0, (S a).HasDensity x {p | p.Prime}

theorem artin_primitive_roots.parts.ii
    (a a_0 b : ℤ) (ha : a = a_0 * b ^ 2)
    (ha' : ∀ n m, m ≠ 1 → a ≠ n ^ m) (ha_0 : Squarefree a_0)
    (ha_0' : ¬a_0 ≡ 1 [ZMOD 4]) :
    (S a).HasDensity ArtinConstant {p | p.Prime}

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one — Artin's Conjecture on Primitive Roots, second half, power version If a = b^m is a perfect odd power of a number b whose squarefree part b_0not≡ 1 pmod4, then…
• artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one — Artin's Conjecture on Primitive Roots, second half, power version If a = b^m is a perfect power of a number b whose squarefree part b_0≡ 1 pmod4, then the…

References

• Wikipedia
• A85397
• LMS14 Lenstra, H.W. et al. "Character sums for primitive root densities" _arXiv:1112.4816_ [math.NT] (2014).
• [Ho67] Hooley, C. "On Artin's conjecture." _Journal für die reine und angewandte Mathematik_ 225 (1967): 209-220.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.