Heath-Brown: Artin Primitive Roots Conjecture

Artin primitive roots conjecture is about the number of primes which have a given fixed integer $a$ as a primitive root. In fact, if the integer is not a square and not equal to $-1$ , we have the following conjecture proved conditionally under GRH (Hooley). (See Hooley’s Proof)

If $a \neq-1 .$ or a perfect square, then there is a constant $C(a)>0 .$ such that

$\displaystyle \left|\left \{p \leq x: a \text{ is a primitive root modulo } p\right\} \right| \sim C(a) \frac{x}{\log x} .$

For $a=2$ , the constant $C(2)$ is called Artin’s constant.

$\displaystyle C_{\text {Artin }}=\prod_{p \text { prime }}\left(1-\frac{1}{p(p-1)}\right)=0.3739558136 \ldots$

This constant is $C(a)$ is the same for a lot of values $a .$

For more general $a$ , write $a= a_0b^2$ with $a_0$ square-free. Let $h$ be the largest integer such that $a$ is a $h$ -th power.

$\displaystyle C(a)= \begin{cases}C_{\text{Artin}}, & \text { if } a_{0} \not \equiv 1(~\bmod 4) \text { and } h=1 \\ C_{\text{Artin}}\left(1-\prod_{q \mid a_{0}} \frac{1}{1+q-q^{2}}\right), & \text { if } a_{0} \equiv 1(~\bmod 4) \text { and } h=1\end{cases}$

The formula for $h>1$ is

$\displaystyle C(a)=\left\{\begin{aligned} C_{\text{Artin}} \prod_{q \mid h} \frac{q^{2}-2 q}{q^{2}-q-1}, & \text { if } a_{0} \not \equiv 1(~\bmod 4) \\ C_{\text{Artin}} \prod_{q \mid h} \frac{q^{2}-2 q}{q^{2}-q-1}\left(1-\prod_{q \mid a_{0}} \frac{1}{1+q-q^{2}}\right.&\left.\prod_{q \mid\left(a_{0}, h\right)} \frac{q^{2}-q-1}{q-2}\right) & \text { if } a_{0} \equiv 1(~\bmod 4)\end{aligned}\right.$

Where do we these constants come from? Obstructions to $a$ be a primitive root $\mod p$ is the extension of prime $q$ satisfying $p\equiv 1 \mod q$ and $a^{\frac{p-1}{q}} \equiv \mod p$ . Now the probability of having this obstruction at $q$ can be seen as the probability that a prime $p$ splits in some Kummer extension $K_q$ whcih by Chebatorev density equals to one over the degree of $K_q = 1/q(q-1))$ . If all these events for $q$ obstructions are independent we arrive at the main term with the constant $C(a).$ The case $a_0 \neq 1 \mod 4$ has the events independent, for other cases, we need to correct for some local dependencies.

But the statement are not proved unconditionally even in the weaker form that there are infinitely many such primes even for a single $a.$

But the first unconditional results established results showing that at least one of a finite set of numbers are primtive roots for infinitely many primes. (Gupta, Murty, Murty)

Later Heath-Brown improved their results by the use of Chen-type switching trick.

Theorem 1: One of $2,3,5$ is a primitive root $(\bmod p)$ for infinitely many primes $p .$ .

He also proved that almost all primes satisfy the Artin’s conjecture (weak). In fact

Theorem 2: There are at most two primes for which Artin’s conjecture fails.

The main ideas are to use the large sieve to estimate of the set of primes for the group generated by three independent element integers modulo the primes is a small subgroup of $(\mathbb Z/p)^{*}$ . Then using linear sieve and the bilinear error terms (along with Bombieri-Friedlander-Iwaniec) to restrict to get a lower bound on number of $p-1$ divisible by only larger primes, to get lower bounds on primes where $p-1$ non-zero residue elements are all generated by the 3 elements.

D. R. Heath-Brown, Artin’s conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) {3 7}(1986), 27-38.

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