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I've heard two conflicting stories on whether or not the Zeta function 100% accurately predicts where primes are.

So does It? Also, is there an error correction formula that makes it 100% accurate if it is not already?

Alethia Arete
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    It's not clear exactly what you are referring to here. There are many relationships between $\zeta$ and the primes. The most direct is given by the Euler product $\zeta(s) = \prod_{p~\text{prime}} (1-1/p^s)^{-1}$ for $\Re s>1$ which shows how the value of $\zeta(s)$ is determined by the location of the primes. Another one is the relationship between the zeros of the $\zeta$-function and the distribution of primes as given by Riemann's explicit formula. – Winther Apr 18 '17 at 17:39
  • Closely related, perhaps a duplicate: Two representations of the prime counting function, beginning with an exact expression for the prime counting function that depends on zeroes of the Riemann zeta function. – hardmath Apr 29 '17 at 15:18

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The Riemann Zeta function is a huge topic in maths, but my understanding is that Schoenfeld (1976) showed that $\zeta$ can put a bound on the error of the prime number theorem with $$| \pi (x) - \text{Li}(x) | < \frac{1}{8\pi} \sqrt{x} \log(x)$$ Where $x \geq 2657$ and with $\pi(x)$ being the prime counting function.

awright96
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