I want to show that for large $n$ the $n$-th prime grows like $n\ln (n)$. Is this correct?
By PNT $$\mathop {\lim }\limits_{x \to \infty } \frac{{\pi (x)\ln (x)}}{x} = 1.$$ Let $x = {p_n}$, so that $\pi (x) = n$, where ${p_n}$ is the $n$-th prime, then $$\mathop {\lim }\limits_{n \to \infty } \frac{{n\ln (n)}}{{{p_n}}} = 1$$ or $${p_n} \sim n\ln (n).$$