$\frac{p_n}{n}$ where $p_n$ is the $n$th prime number?

Question

Is proven that $\frac{p_n}{n}$ diverges where $p_n$ is the $n^{\text{th}}$ prime number?

Answer 1

From the Prime Number Theorem, we know:

$$p_n\sim n\log n$$

Therefore, $\dfrac{p_n}{n} \sim \log n$, and therefore diverges.