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jfribrg
2005-Feb-16, 07:56 PM
Given the spirited postings from the recent math threads, I decided to ask a question I asked a while ago: How does one calculate the complex non-trivial zeroes of the Riemann Zeta function. I've seen various books about the Riemann hypothesis, but they all either assume you know how to do the calculation, or they simply state that the calculation is beyond the scope of the book. I haven't found any web sites that discuss how to do the calculation. Most simply discuss the existence of such zeroes, with maybe an example or to, but no computational issues are addressed.

Severian
2005-Feb-16, 10:44 PM
One doesn't just go and calculate the nontrivial zeroes; the (unproven) Riemann Hypothesis is that they are all on the line Re(z) = 1/2 and it has been shown that they are all in the strip 0 &lt;= Re(z) &lt;= 1 and are symmetric about that line (though I don't know the proof). So no one knows how to calculate all the complex non-trivial zeroes (since if it was known, we'd know if the RH was true).

A Thousand Pardons
2005-Feb-17, 12:22 AM
So no one knows how to calculate all the complex non-trivial zeroes
I think jfribrg was asking how to calculate some of them. :)

After all, from what I read, that's how Riemann twigged to his hypothesis.

Severian
2005-Feb-17, 03:16 AM
Oh yeah, I guess you're right

Honestly, I don't know much about it. I found a few things with some simple minded Google's but I would guess anyone has found those. I did find a paper of Turing's online which seems to calculate some of them at http://www.turingarchive.org/browse.php/B/21 (the reference which led me to it was at http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html) .

Sorry, though, I don't know enough number theory or complex analysis to tell you much.