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I watched a video explaining that using the non-trivial zeros of the Riemann zeta function, we can approximate the prime counting function to have a pretty high accuracy

I would like to know, how many zeros we must have (find) to obtain all primes less than 100 for example ? or this doesn't work like that ? if so what can we benefit from two or three zeros (let's say the first ones) ?

Also, why is this a "high accuracy", and not exact result ?

video link time 15:45 : https://youtu.be/dktH8hJadyU?t=948

Billy senders
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    https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions#Riemann's_explicit_formula – Qiaochu Yuan Oct 02 '20 at 00:00
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    The guaranteed magnitude of the error term $|Li(x)-\pi(x)|$ is dramtically smaller if the Riemann hypothesis is true, but still there is no direct way to determine $\pi(10^{300})$ or the $10^{300}$ th prime. – Peter Oct 02 '20 at 07:16
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    There are recursive methods to determine $\pi(x)$ for which we arrived around $10^{26}$ (the exponent in the current record might be somewhat smaller or larger) , but they neither allow calculating exact values for huge $x$. In particular, we cannot determine huge primes this way. – Peter Oct 02 '20 at 07:16
  • thanks @Peter, so if we discover a new way that allows calculating exact values for huge x, a direct way to determine π(10^300) or the 10^300 th prime, we may say that the Riemann hypothesis is no longer interesting ? – Billy senders Oct 02 '20 at 22:30
  • No, the riemann hypothesis is a very important conjecture, it would be very useful to prove it. When I googled "riemann hypothesis" , the first (!) hit was a claim that it has been proven - quite sad. Why people over and over again claim such things, no idea. – Peter Oct 03 '20 at 07:51
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    To obtain $\pi(x)$ using the zeta zeros see the papers from this thread (around $70$ billions zeros were needed to compute $\pi(10^{24});$). – Raymond Manzoni Oct 06 '20 at 18:33

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