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  1. How many contiguous zeros of zeta are known, to what height
  2. How many contiguous primes are known, to what height
  3. How many zeta zeros determine how many primes, to what exactness

For example, would knowing the first 1,000 zeta zeros pinpoint the location of the first 1,000 primes, exactly? (Assuming all zeros are on the critical line, which found ones appear to be)

Is there a formula that would match a certain number of zeta zeros to a certain number of primes that are determined. Or perhaps, could you calculate more than would be directly calculated by assuming RH, and find more primes. Would even the first 5 roots of zeta give any information on a large number of primes, etc.

I know these questions are very general, I sense that many zeta zeroes would need to be calculated to even find the first thousand primes roundable to their integer values, and even then some might round the wrong way?

PGH

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    $\begingroup$ I can’t quite imagine how could a sensible answer to question 2 look like. What does it mean to “know” a prime? For one thing, the list of all primes in a given interval can be computed on demand faster than one could read them from a disk. $\endgroup$
    – Emil Jeřábek
    Commented Jul 2, 2014 at 15:25
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    $\begingroup$ I once asked Lenstra what was the largest prime that he actually knew all the digits of. Of course he had an answer: the repunit (10^1031-1)/9. $\endgroup$
    – Stopple
    Commented Jul 7, 2014 at 15:50

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You would need to know all the Riemann zeros to determine a single prime exactly, and vice versa. The primes and the zeros are on the opposite sides of a Fourier transform in Riemann's Explicit Formula, so the mathematical version of the Heisenberg Uncertainty Principle applies:

http://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle

For the approximate values of small primes as determined by low-lying zeros, you might enjoy looking at http://www.math.ucsb.edu/~stopple/explicit.html

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    $\begingroup$ But a prime number is also an integer. So knowing it to accuracy $1/2$ would be enough to pinpoint it, and this could be done using about $p$ zeros to determine $p$. (The last para of the question indicates that this would be acceptable.) $\endgroup$
    – Lucia
    Commented Jul 2, 2014 at 15:05
  • $\begingroup$ Thanks. My concern is that those off by about 1/2 would sometimes round the wrong way, but then again, that would be an even number, naturally. $\endgroup$
    – Paul Hjelmstad
    Commented Jul 9, 2014 at 20:52

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