Let $p$ be a prime number. I am interested in knowing how many primitive roots mod $p$ there are; at least, gaining some insight into the distribution of primitive roots mod $p$.
If I need to go looking for a primitive root, how far down the list of integers should I expect to look before I find one?
I know there are $\phi(p-1)$-many primitive roots mod $p$. Therefore the ratio of primitive roots mod $p$ is given by $\phi(p-1)/(p-1)$.
I appreciate that there are infinitely many primes and that the behaviour of the first 100000 need not tell us anything about the overall behaviour. That being said, I am hoping someone could explain some of the features of this plot that stand out to me. For example:
The Number of primitive roots is bounded between 1/5 and 1/2. It seems some might sneak lower than 1/5.
There are a number of dense lines. For example: It seems there are lots of primes with 1/3 of integers being primitive roots.
If anyone can point out any references about the distribution of primitive roots. Or say anything about what might be happening here, that would be great.