In https://www.quora.com/What-is-the-relationship-between-the-Riemann-Hypothesis-and-prime-numbers and https://en.wikipedia.org/wiki/Prime-counting_function#Exact_form talk of exact formulas for the prime number counting function (assuming the Riemann Hypothesis and probably above some value) , whereas https://en.wikipedia.org/wiki/Riemann_hypothesis#Distribution_of_prime_numbers talks of a best bound for it. So, which is it: exact or not? That is, if we assume the RH is true and we are given a large x, can we calculate exactly pi(x) or can we still only approximate it? (I understand that without the RH, we can only approximate it.)

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    $\begingroup$ Note that the exact formula on WP (which is unfortunately guarded by a "citation needed") asks you to evaluate a $\sum_\rho$, i.e., you need to run over all non-trivial zeroes. One would still need to determine after how many summands the error is $<\frac12$. $\endgroup$ Feb 9, 2019 at 9:15
  • $\begingroup$ Of course we can calculate $\pi(x)$ exactly without the Riemann Hypothesis. It's just a question of testing every integer up to $x$, and keeping count of the number of primes. And there are more efficient methods. RH would give us another way to calculate $\pi(x)$ exactly, though it's not clear to me that it would be more efficient than the methods that don't use RH. $\endgroup$ Feb 9, 2019 at 10:02
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    $\begingroup$ The references provided here may help ($\pi(n)$ was evaluated up to $n=10^{24}$ with a bounded error using $70$ billions non-trivial zeros). $n=10^{25}$ was reached but it seems that the Meissel-Lehmer method took the lead again after being blocked at $10^{23}$... $\endgroup$ Feb 9, 2019 at 10:05


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