Let $f$ be a following function :
${ f: \mathbb{N} \rightarrow \mathbb{P} }$
${ f(n) = p_n }$, where $p_n$ is n-th prime number.
I want to prove or find a proof that ${ f \in \Theta\left(n\log\left(n\right)\right) }$ (using Big-Theta notation) which seems correct by empirical evidence: https://www.desmos.com/calculator/ylpnljuh1y
If I assume that ${ \Pi \in \Theta(\frac{x}{ln{x}}) }$, where $\Pi$ is a prime-counting function it seems me easy to prove because inverse of ${n\log{n}}$ is ${\frac{x}{W(x)}}$, where $W$ is lambert W-function. I can prove that ${\frac{x}{W(x)} \in \Theta(\frac{x}{ln{x}}) }$ but I can't find proof or prove that ${ \Pi \in \Theta(\frac{x}{ln{x}}) }$. Can anyone help me?
I apologize for not being word-perfect in English.