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While looking for information about the Riemann zeta function, I kept running into the claim that there is an algorithm to decide whether or not a zero of the function is off the half-line. Is this true, or is there only a semi-decision procedure?

I could picture a situation where we find a spot where it appears that the Riemann zeta function has a repeated zero, but we just can't decide whether or not the zeros are at the same place.

On the other hand, the Riemann hypothesis can be expressed as a $\Pi_1^0$ statement in PA, and perhaps there is a way to translate this into an algorithm for deciding (up to any finite height) whether zeros are on the half-line or not.

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    $\begingroup$ My vague recollection is that this is known to be decidable under the conjecture that all zeros of zeta are simple - as you indicate, if we had a double zero, then there isn't an obvious way to tell with certainty that they do lie on the line. Of course if the zero is simple, then it must lie exactly on the line. $\endgroup$
    – Wojowu
    Commented May 19, 2021 at 20:27
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    $\begingroup$ The proof I know that RH is $\Pi_1^0$, which involves expressing it as an explicit inequality about primes, pretty clearly does not imply any algorithm about any particular zero or zeroes up to a finite hight, because the size of the error in the prime-counting function depends continuously on the distance of the zero from the line, so if it's very close you will need many primes to see it. $\endgroup$
    – Will Sawin
    Commented May 19, 2021 at 21:32
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    $\begingroup$ That RH is equivalent to $\Pi^0_1$ statement follows from semidecidability of this problem (if there is a zero off the line, we can computationally verify it). So I don't think this method can be translated to verify that this is something that is in fact decidable. $\endgroup$
    – Wojowu
    Commented May 19, 2021 at 22:36
  • $\begingroup$ Aran Nayebi, On the Riemann hypothesis and Hilbert's tenth problem, presents an overview of the methods that prove the Riemann hypothesis is a $\Pi^0_1$ sentence. web.stanford.edu/~anayebi/projects/RH_Diophantine.pdf See also mathoverflow.net/questions/31846/… $\endgroup$
    – Gerry Myerson
    Commented May 20, 2021 at 7:00

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