Auch wir sind überrascht wie erfolgreich diese Auswanderersendungen geworden sind und freuen uns natürlich sehr, dass die Familie Reimann in Texas Fuss fassen konnte und wünschen Ihnen weiterhin viel Erfolg - in den USA und im deutschen Fernsehen.

© Flavia Westerwelle

It centers on the Riemann Zeta Function, given by

$latex \zeta (s) = \sum_{i = 1}^\infty \frac{1}{i^s} &s=1$

The function being defined over the complex plane. The Hypothesis is, in its most basic form, that if $latex \zeta (s) = 0$, then s = 1/2 + t*i, for some real number t. That any zero of the zeta function has a real part of one half.

Why this is interesting though is the connection with prime numbers. There's a function called the prime counting function, pi(x), and it basically gives the number of prime numbers between 0 and n. As the distribution of primes is so bizarre, being able to calculate pi(x) correctly, without just counting all the prime numbers from 1 to x, would be a very neat trick. We have some very nice approximations for pi(x), involving integrals of relatively simple functions, but the question is the, how good are these approximations?

So enters the Riemann Hypothesis. Basically, it's been shown that the Riemann Hypothesis, as stated above, is equivalent to saying, for our best approximation of pi(x), pi(x) differs from this approximation by no more than some constant times $latex \sqrt{x}*ln(x)$, for x sufficiently large.

Which is a very interesting result.

But a proof was proposed a few months ago, and it seems really quite promising, and at the very least opens up some new avenues for further inquiry. As usual for such things, I don't have the background to explain fully, but I think I can break down a couple of the key points to an understandable level. Maybe Mark can pick this up?

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