Number of natural numbers that have primitive roots

[mathmari]

Hey! :giggle:

How can we calculate the number of natural numbers between $2$ and $n$ that have primitive roots?

Let $m$ be a positive integer.
Then $g$ is a primitive root modulo $m$, with $(g,m)=1$, if the modulo of $g\in (Z/m)^{\star}$ is a generator of the group.

We have that $g$ is a primitive root modulo $m$ if it is a generator of a group, i.e. $m$ has a primitive root if $\mathbb{Z}_m$ is cyclic, right?

$\mathbb{Z}_m$ is cyclic if $m=1,2,4$ or $m=p^k$ or $m=2\cdot p^k$ for $p$ prime.

That means that the number of natural numbers that have a primitive root is $\#\{1,2,4,p^k, 2\cdot p^k\}$ for $p$ prime.

So we have to calculate the number of primes between $2$ and $n^{\frac{1}{k}}$ to calculate then the number of elements of the form $p^k$ and $2\cdot p^k$.

Have I understood that correctly? :unsure:

 

[I like Serena]

Looks about right. :unsure:

 

[mathmari]

Looks about right. :unsure:

To calculate the number of these primes do we use the density of primes? :unsure:

 

[I like Serena]

To calculate the number of these primes do we use the density of primes?

We can only approximate the density of primes.
So we cannot use it to find an actual number.
Assuming that we want a 'hard' number, I think we should express it in terms like 'the number of primes between $2$ and $n$'. :unsure:

 

[mathmari]

We can only approximate the density of primes.
So we cannot use it to find an actual number.
Assuming that we want a 'hard' number, I think we should express it in terms like 'the number of primes between $2$ and $n$'. :unsure:

Actually I want to calculate the limit $\displaystyle{\lim_{n\rightarrow \infty}\frac{a_n}{n}}$ where $a_n$ is the above number. So do we need the actual number to calculate this limit? :unsure:

 

[I like Serena]

Actually I want to calculate the limit $\displaystyle{\lim_{n\rightarrow \infty}\frac{a_n}{n}}$ where $a_n$ is the above number. So do we need the actual number to calculate this limit?

No. I think we can use the density of primes to calculate that limit. :unsure: