# Riemann Hypothesis implies Prime number theorem?

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)

• What do you mean ? A known theorem can be the consequence of the truth of an unsolved conjecture. What is your question ? Jun 24 at 19:35
• If I remember correctly, the prime number theorem is equivalent to the fact that the nontrivial zeros of the zeta function lie in the interior of the critical strip, i.e., they satisfy $0<\text{Im}(z)<1$. The Riemann hypothesis says they satisfy $\text{Im}(z)=\frac12$. Jun 24 at 20:33

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