withoutaface said:
If a computer found a root that didn't lie on the strip you could just sub it in manually and prove it that way, couldn't you?
It is known that they all lie in the critical strip. The Riemann Hypothesis says that all the nontrivial zeros of ζ(s) lie on the critical
line, Re(s)=1/2. If that's what you meant to say, then theoretically, yes.
But you're a braver man than me if you try to do it by hand. Not even Odlyzko himself would do that. It would be a huge number. Incidently, Riemann himself did calculate the first few zeros by hand and I've seen the page from the Nachlass on which he did the first one. It's quite a mess, and very hard to understand. And that value was only pretty small (0.5±i(14.134...)). Subsequently, further calculations by hand were done by Gram, Backlund and Hutchinson, but thereafter all the work was done by computers.
Gram, Backlund and Hutchinson used Euler-Maclaurin summation. But Riemann's method was different. He used the Riemann-Siegel formula. This formula is what most of the computer work is based on, until very recently when Gourdon used the Odlyzko-Schönhage algorithm (a more powerful generalisation of the Riemann-Siegel formula) to calculate the first 10 trillion zeros, and found they are all on the line.
Hardy proved there are infinitely many zeros on the line, so computers probably can't be used to actually
prove the Riemann Hypothesis. Nevertheless, they might be able to be used to
disprove it if they find one off the line.
But of course no_arg won't have a bar of it because it's all done by computers thesedays. The best anyone has done by hand, was by Hutchinson, and he proved by hand that the first 138 are on the line. Compare that with 10 trillion! Give me computers any day! I believe them! Especially because no_arg doesn't!
