There is a result that if $p_n$ is the $n$th prime, then $p_n\sim n\log n$ as $n\rightarrow\infty$.
I wonder: Is it a direct consequence of the prime number theorem $\pi(x)\sim x/\log x$? The theorem says that there are approximately $n/\log n$ primes less than or equal to $n$. So there are approximately $n$ primes less than or equal to $n\log n$. So $p_n$ is approximately $n\log n$.
But I'm having trouble turning this into a formal argument. We have $\lim_{n\rightarrow\infty}\dfrac{\pi(n)\log n}{n}=1$. How would it show that $\lim_{n\rightarrow\infty}\dfrac{p_n}{n\log n}=1$?
Best Answer
The proposals are $p_n\sim n\log n$ and $\pi(x)\sim \frac{x}{\log x}$. Consider:
$$p_{\pi(x)}\sim \pi(x)\log\pi(x)\sim\frac{x}{\log x}\log\left(\frac{x}{\log x}\right)=\frac{x}{\log x}\left(\log x-\log\log x\right)\sim x$$
and
$$\pi(p_n)\sim\pi(n\log n)\sim\frac{n\log n}{\log(n\log n)}=\frac{n\log n}{\log n+\log\log n}\sim n$$
Try formailizing this (giving more meaning to $\sim$) to show that $p_{({.})}$ and $\pi({.})$ are asymptotic inverses.
To formalize more: Let $f(x)=\frac{x}{\log{x}}$ and $g(x)=x\log{x}$. Then $$\frac{f(g(x))}{x}=\frac{1}{x}\frac{x\log{x}}{\log(x\log{x})}=\frac{1}{1+\frac{\log\log x}{\log x}}\xrightarrow{x\to\infty}1$$
And $$\frac{g(f(x))}{x}=\frac{1}{x}\frac{x}{\log{x}}\log\left(\frac{x}{\log{x}}\right)=\frac{1-\frac{\log\log x}{\log x}}{1}\xrightarrow{x\to\infty}1$$
And so for large $x$, $f(g(x))\sim x$ and $g(f(x))\sim x$, where $A\sim B$ means that their ratio approaches $1$.
All this is to say that $f$ and $g$ are asymptotic inverses. So if we start with the premise that the $n$th prime number $p_n$ is approximately $g(n)$ (in the same "ratio approaches 1" sense), and then define $\pi(x)$ as the inverse concept (the number of primes less than a given $x$), then $\pi(x)$ is approximately $f(x)$ (again in that sense).