It is well known that many problems in number theory have an analogue on the ring of polynomials over finite fields and vice versa, the primes in $\mathbb{F}_q[x]$ being the irreducible polynomials. For instance, the number of primes up to $x$ is $\sim x/ \log x$ by the Prime Number Theorem, while the number of irreducible polynomials of degree $n$ over $\mathbb{F}_q$ is $\sim q^n/n$. Note this is $N/ \log_q N$, where $N$ is the number of monic polynomials of degree $n$ over $\mathbb{F}_q$.
I wonder what are, if any, the numbers which are analogous to primitive polynomials (irreducibles of degree $n$ whose roots are primitive elements in $\mathbb{F}_{q^n}$) over finite fields?
Since the number of primitive polynomials of degree $n$ over $\mathbb{F}_q$ is $\phi(q^n-1)/n$, I suspect that the amount of such numbers below $x$ is $\phi(x)/ \log x = \dfrac{\phi(x)}{x} \dfrac{x}{\log x}$ if we assume that $x \in \mathbb{N}$. Note the $x / \log x$ of the PNT while $\phi(x)/x$ is the proportion of integers below $x$ which are coprime to $x$. Thanks!