Riemann Hypothesis implies Prime number theorem?

Question

Following this thread Prime number theorem and Möbius μ function

I was looking into Apostol's book, Introduction to Analytic Number Theory, Springer 2000. And also note that RH is equivalent to Mertens conjecture, I thought if one can prove PNT using RH (Given the hypothesis is true)

Answer 1

Yes you can. See: https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences

Some simple thoughts: The simplest and obvious connection i could think between something similar to Riemann zeta function and prime number estimates is: $$\sum_n \frac{1}{n} = \sum_p \frac{1}{p} \times \sum_p \frac{1}{p^2} \times \sum_p \frac{1}{p^3} \times ..... $$

$$\sum_p \frac{1}{p^2} \times \sum_p \frac{1}{p^3} \times ..... \leq \left(\sum_n \frac{1}{n^2}\right)^2$$

Hence:

$$\sum_p \frac{1}{p}$$ diverges. Hence asymptotically $n^{th}$ prime number $p_n$ is such that for every $\epsilon>0$ the following happens infinitely often, $$n < p_n < n^{1+\epsilon}$$

Of course there are clearly better estimates of $n^{th}$ prime number than this. But i see this as the easiest connection.