I am reading a survey on "Artin's Primitive root Conjecture" and I am having a hard time understanding some statements made by the author.
- The author states that, The prime number theorem in the form $$\pi(x) := \#\{p\le x\} \sim Li(x), x \rightarrow \infty $$ suggests that the probability that a number is prime is $\frac{1}{\log(n)}$. I think this probability arrives from $\frac{Li(x)}{x}$. But, I guess the statement should have been *probability that a number not greater than n is a prime is $\frac{1}{\log(x)}$. Kindly, explain this
- Next, the author states that We expect $\displaystyle \sum_{2\le n\le x }\frac{1}{\log(n)} \le x $ and asymptotically this is equal to $Li(x)$. Thus from both $n$ and $4n+1$ to be prime and $n \equiv 2 (\text{ mod } 5)$ we expect a probability of $$\frac{1}{5\log^2(n)}$$ Assuming independence of these three events. The author further claims that since these three events are not independent we have to correct by some positive constant $c$ and hence that up to $x$ there are atleast $\displaystyle \frac{cx}{\log^2(x)}$ ( note that $\displaystyle \sum_{2 \le n \le x} \frac{1}{\log^2 n} \sim \frac{x}{\log^2(x)})$ primes $p$ such that $10$ is a primitive root mod $p$. Hence, we expect there are infinitely many primes $p$ having $10$ as a primitive root mod $p$.
I have no idea how the author arrived at this statement and I am not able to find any relevant resource to understand the claim made by the author.
Reference: Artin's primitive root conjecture, a survey - Pieter Moree,