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sunday entertainment: the Riemann Hypothesis


Working on some stuff on n-dimensional topologies I remembered the Riemann Hypothesis. I'm not sure when I first read about it... but it keeps popping up in my head (simply because it's a challenge--not that I have any delusions that I can actually solve it! :)). So as a way of relaxing a bit from work, here's some background on that. Not that it's something useful, but it should be at least entertaining.

A couple of years ago, a $1,000,000 prize was created for whoever came up with the proof. What I didn't know was that they actually had seven "millenium prize problems" of which the Riemann Hypothesis was one. Here is the list of all seven problems. One million per proof. Not bad!

Going back to the Riemann Hypothesis. Proving it would be interesting for number theory and the distribution of prime numbers, but it would have no direct practical applications whatsoever. Otherwise, I can see the headlines (maybe for The Onion?):

UN Security Council Declares Understanding the Distribution of Prime Numbers is Priority One - Troops To Be Deployed in the Real Region of C with R > 1 - President declares "we will hunt down these prime number folks and we will characterize them. If you ask me, there's something evil about a number being divisible only by itself and one."
Joking aside, the new mathematical techniques that usually have to be invented to solve these open problems do have applicability. But I digress.

The Hypothesis is generally considered one of the most important unproven hypothesis in mathematics, and I've read somewhere that some people think it was even more important than Fermat's famous Last Theorem. To put things in context, Fermat's Last Theorem stated that given:

xn + yn = zn

there are no integer solutions for n > 2 and x,y,z != 0. The statement of Fermat's Last Theorem is relatively simple and self-contained. the Riemann Hypothesis is not. Consider, from the prize page:
The Riemann hypothesis is that all nontrivial zeros of the Riemann zeta function have a real part of 1/2
Sure it reads easier than Fermat's theorem, but it gets away with that by putting all the complexity in the definition of the Riemann zeta function.

What I find it fascinating how you actually get to it. After some reading, it would appear to go like this: Riemann was trying to derive a formula that would calculate the number of primes lower than a given boundary number n. In doing this, he started to look at an infinite series based on a complex number, s. That series is defined by the Riemann zeta function:

zetafunction.png

which converges. Okay, given that function, the Riemann Hypothesis is saying that all the nontrivial zeros exist only at values of s that have a Real component of 1/2.

And this is one of the things that I find great about mathematics: you start pulling out threads and in the end you are left with just one or two hypothesis. If you prove those, everything else falls into place, domino-style. And what you're actually proving often appears to have no relationship at all with what you were originally interested in!

For more there's this cool page in MathWorld with a lot of interesting information on the Hypothesis and some of the attempts to solve it so far.

As I said, maybe not useful, but at least entertaining. :)

Categories: science
Posted by diego on September 28 2003 at 3:56 PM

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