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Riemann Hypothesis - Numberphile
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[Prof Frenkel] Can I ask you a question Brady?
[Prof Frenkel] What is the most difficult way to earn a million dollars?
[Brady] Making Youtube videos. [Prof Frenkel] *laughs*
[Prof Frenkel] Well, you probably know much more about that than I do.
[Prof Frenkel] One of the most difficult ones is to solve one of the Millenium Problems in Mathematics,
which were set by the Clay Mathematical Institute in the year 2000.
One of these problems is called "The Riemann Hypothesis".
It refers to a work of a german mathematician, Bernard Riemann,
which he did in the year 1859.
This is just one of the problems. In fact, there are seven.
And one of them has been solved so far.
And interestingly enough, the person who solved the problem has declined the one million dollars.
So...
It just shows that mathematicians work on these problems, not because they want to make some money.
I think it is now the most famous problem in mathematics.
It took the place of Fermat's Last Theorem,
which was solved by Andrew Wiles and Richard Taylor in the mid-1990s.
[Brady] But that wasn't a Millenium Problem. [Prof Frenkel] That was not a Millenium Problem.
[Prof Frenkel] The most essential thing here is what we call the Riemann Zeta function.
And the Riemann Zeta function is a function, so ...
A function is a rule which assigns to every value some other number.
And the Riemann Zeta function assigns a certain number to any value of s,
and that number is given by the following series:
1 divided by 1 to the power of s,
plus 1 divided by 2 to the power of s,
plus 1 divided by 3 to the power of s,
4 to the s, and so on.
So, for example, if we set x = 2.
Zeta(2) is going to be 1 divided by 1 squared plus
1 divided by 2 squared,
plus 1 divided by 3 squared,
plus 1 divided by 4 squared,
and so on.
So, what is this?
This is one.
This is 1 over 4.
This is 1 over 9.
1 over 16...
So this is an example of what mathematicians call a convergent series,
which means that, if you sum up the first n terms,
you will get an answer which will get closer and closer to some number.
And that number to which it approximates is called the limit.
But the limit here is actually very interesting.
And it has been a famous problem in mathematics to find that limit.
It is called the Basel problem,
named after the city of Basel in Switzerland.
And this Basel problem was solved by a great mathematician: Leonhard Euler.
And the answer is very surprising.
What Euler showed is that this sums up to pi squared over 6.
So you may be wondering.
What does this sum has to do with a circle?
Why would pi squared show up?
But Euler came up with a beautiful proof.
I'm not going to explain it now, but it's something that you can easily find online.
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