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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)

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    $\begingroup$ What do you mean ? A known theorem can be the consequence of the truth of an unsolved conjecture. What is your question ? $\endgroup$
    – Peter
    Jun 24 at 19:35
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    $\begingroup$ 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 $\text{Im}(z)=\frac12$. $\endgroup$
    – Andreas Blass
    Jun 24 at 20:33

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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.

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