The Riemann Hypothesis





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Published on Jan 17, 2014

The Riemann Hypothesis is one of the Millennium Prize Problems and has something to do with primes. What's that all about? Rather than another hand-wavy explanation, I've tried to put in some details here. Some grown-up maths follows.

More information: http://www.claymath.org/publications/...


CORRECTION: The functional form of the zeta function is a reflection around the point 0.5, not the line x=0.5. But you can think of that as a reflection in the x-axis followed by a reflection in the line x=0.5.

For example, if we start with zeta(x + iy) and reflect it in the x-axis we get zeta(x - iy). Reflect that in the x=1/2 line we get (junk)*zeta(1 - x - iy). And reflect that in the x-axis again we get (junk)*zeta(1 - x + iy). When zeta(x + iy) = 0 so are all the others.

The zeros then look like this http://en.wikipedia.org/wiki/File:Zer... That's four zeros for the price of one.

The zeros are still symmetric around the x=1/2 line. The reflection of zeta(x + iy) is (junk)*zeta(1 - x + iy). In the video I mixed up zeta(1 - x + iy) with zeta(1 - x - iy).

But the point is the same, the error in the Prime Number Theorem is minimised if all the zeros lie on the line x=1/2.


Some people have asked about my final claim that you don't get the prize for a counterexample. Here is the rule from the Clay Institute:

"if a counterexample is proposed, the SAB will consider this counterexample after publication and the same two-year waiting period as for a proposed solution will apply. If, in the opinion of the SAB, the counterexample effectively resolves the problem then the SAB may recommend the award of the Prize. If the counterexample shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author. The money for this prize will not be taken from the Millennium Prize Problem fund, but from other CMI funds."

In other words, there are no guarantees either way.


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