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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.

  • This series is just one example of this Riemann Zeta function,

  • but you can try to do the same for any other value of s.

  • So, for example, if you take s=3, you will get the reciprocals of all the cubes,

  • and you sum them up, and so on.

  • So this will, again, be a convergent series, and you can wonder what that answer is.

  • That would be zeta(3).

  • You can also try to substitute negative numbers.

  • And this is very interesting, because if you substitute... if you just substitute

  • If s = -1, then what are we going to get?

  • So you will get 1 divided by...

  • 1 to the 1 to the -1,

  • plus 1 to the 2 to the -1,

  • plus 1 over 3 to the -1...

  • If you take the reciprocal of something, which is the inverse of something,

  • then you will get that thing.

  • So this will be 1,

  • this will be 2,

  • this will be 3,

  • this will be 4...

  • [Prof Frenkel] Does it look familiar? [Brady] Yes, I have seen that before.

  • [Prof Frenkel] We have arrived at the famous sum of all natural numbers:

  • 1 + 2 + 3 + 4 ...

  • But, you see, now we obtained in the context of the zeta function.

  • So this is what we call a divergent series.

  • There is no obvious way how we could possibly assign a finite value to it.

  • This sum is infinite, it does not converge to any finite value.

  • But, this context...

  • If we put this value, this infinite sum, in the context of this function,

  • there is actually a way to assign a value to s=-1.

  • And this is what Riemann explained in his paper.

  • And so what Riemann said is that, actually, we should allow s to be, not just a natural number

  • for example, 2, or 3, or 4, when the series is convergent

  • but we should allow also all possible real numbers.

  • And not only real numbers, but also complex numbers.

  • The way you get complex numbers is by realizing that, within real numbers,

  • you cannot find the square root of -1.

  • Then what to do?

  • One way is to ban the square root of -1 and say,

  • "This doesn't exist, we cannot use it"

  • But, in mathematics, we have understood, a long time ago, that actually there is a much better way to treat this.

  • the square root of -1.

  • We can simply adjoin it to the real numbers.

  • Think of real numbers as points on a line.

  • Here is 0,

  • and here is 1,

  • and here is 2,

  • and then you can mark your favorite fractions.

  • For example, one half is exactly in the middle way between 0 and 1.

  • And say,

  • 1 1/3 would be a third of the way between 1 and 2.

  • But then, you also have things like square root of 2...

  • For example, somewhere here.

  • And then, there is pi, which is just to the right of 3.

  • So all the real numbers live here.

  • Square root of -1 cannot be found anywhere on this line.

  • But we don't give up. We say, "You know what?"

  • "Let's actually draw a plane, let's draw another coordinate system"

  • "And let's mark square root of -1 on this new coordinate axis."

  • You see, if we do that, then every point on this plane becomes a number.

  • So that would be 2 times square root of -1,

  • 3 times the square root of -1, ...

  • But more than that, let me find a number which is on the intersection of this line.

  • I can draw a vertical line which goes from 2

  • and I can go... can also draw a horizontal line.

  • Then there's this point of intersection.

  • So this point also would represent a number,

  • which would be 2 plus 3 times square root of -1.

  • So, in other words, a general number is going to have what we call a real part,

  • that is the projection onto this axis;

  • and the imaginary part, that's the projection on the vertical one.

  • The notation is a little bit clumsy.

  • Instead of square root of -1, they write i.

  • So then for example: instead of writing 2 + 3 square root of -1, we'll just write 2 plus 3i.

  • It's an imaginary number, we imagine it. We cannot find it on this real line.

  • So we have imagined it, and then we have adjoined it in our imagination.

  • Real numbers comprise all points on the real line, on this axis;

  • and complex numbers comprise all the points on this brown paper,

  • if you could extend the brown paper all the way to infinity.

  • Right?

  • So let's go back to Riemann.

  • What Riemann's insight was is he said "look, let's think of this argument of the Zeta function, this number s...

  • Initially, we thought that s could be 2, 3, 4, and so on.

  • But then we realised that actually any real number to the right of number 1...

  • not including number 1, because actually in this case you cannot assign a value, it's a divergent series,

  • so it goes to infinity.

  • But anything to the right, and then drawing and marking it with red...

  • For all of them, this function is actually well defined.

  • So... But then he said '"We can actually do more... We can think of s as being a complex number."

  • So instead of thinking of s as just being a point on this line, we can take s anywhere.

  • It will be convergent if it is to the right of this line.

  • So you see if this is the line, which sort of to the right of this line... live all the complex numbers

  • whose real part is greater than 1.

  • So, it turns outand it's very easy to showthat anywhere in the shaded area, except for this line

  • so to the right of this line.

  • Now, for any value of s in this area, this function converges to something.

  • So if I put 6 + 9i into the Riemann Zeta function, I'll get a convergent series...?

  • That's right. You get a convergent series. It will converge to something,

  • which is not going to be a real number.

  • It's goint to be a complex number, because you're going to add up infinitely many complex numbers.

  • But there will be a certain number to whichwhich will be a closer and closer approximated

  • as you go along summing up the series.

  • So far, basically everything to the right of this line... —Gives us a bona fide value

  • will come out to play. —will come out to play, and will give us a bona fide value.

  • Can the imaginary part go in negative?

  • Yeh, but the imaginary part... yes. The imaginary part is okaycan be negative or positive.

  • But the real part has to be greater than 1.

  • But, now, you are in the context of a theory of... functions with complex arguments.

  • And it is what we call a holomorphic function.

  • So it has some very special, very nice properties.

  • So one of the properties that this kind ofwhat we call holomorphicfunctions enjoy is what we call

  • analytic continuation. So we can extend the definition, i.e. the domain of definition of the function.

  • There are methods which allowwhich enable usto kind of push the boundary,

  • and kind of... go and expand the domain in which the function is defined.

  • And in his seminal paper, Riemann did precisely that.

  • He explained how to extend this function to all possible values, except for one.

  • So there's only one value where there is nothing you can do;

  • and somehow it will be undefined.

  • And this is what we call a 'pole' or a 'singularity'.

  • And what is that value? That value is actually s = 1.

  • So this is somehow... this is a bad point in some sense.

  • This is a point where we cannot extend....

  • And i and 0 are components? There's no imaginary components.

  • That's right. So this is a point which is actually a real number.

  • So it's funny, because you would think that 1 is in some sense easier and better than i.

  • But, at i, this function will be perfectly well defined.

  • But at 1 it will not be well defined; it will have a singularity.

  • But, luckily, it's the only singularity.

  • So, what it means in particular, is that there is a way to assign a value to -1

  • In other words, there is a value Zeta of -1,

  • where by Zeta we now mean this extended function.

  • A function analytically continued to the whole complex plane.

  • And that value will beyou guessed it — -1/12.

  • Okay.

  • It is in this sense that people say that you could regularize the sum 1+2+3+4...,

  • and assign to it the value -1/12. Because it shows up as the value of the zeta function where, naively,

  • you are supposed to get 1+2+3+4...

  • But, now, you are getting this value by a much more sophisticated procedure.

  • By starting with a function of a complex argument, and extending it beyond the initial domain of definition.

  • So no matter what number I pluck from here or here, or here, or here or here, and feed it into the function,

  • I'll get a number of some sort...

  • You will get a well-defined numberuniquely defined number.

  • And you can calculate it on a computer, because this number can be represented by some integral for example.

  • There is a explicit formula for it. Okay? So... and...

  • you will calculate, I will calculatewill get the same result. There is no ambiguity.

  • The only point where it's not well defined is the point s=1.

  • This is like the Achilles' heel or the weak... —It's an Achilles' heelthat's a very good way to put it.

  • It's an Achilles' heel of that function.

  • But it's very important; it's responsible for a lot of things that's happening to it.

  • So it's a very important point.

  • So Riemann's hypothesis is the following:

  • it's about the zeros of the zeta function.

  • In other words, it's a question aboutfor what valuesfor which s we have zeta(s) = 0.

  • That is the one million dollar question.

  • For which value of s do we havedoes this function equal 0.

  • And so... Riemann...

  • Here there is one point that one has to make, which is that

  • there are some sort of obvious zeros.

  • So it just so happens that the value at -2, for example, is equal to 0.

  • -4 is equal to 0.

  • So, in other words, all the even negative numbers, for some reasonit just so happens

  • and you can see it for example from the function equation easily

  • that the value is going to be 0.

  • So there's some obvious zeros so to speak, which we already know.

  • The question is what elsewhere else are the zeroswhere are there other zeros...?

  • And it's actually very easy to see... that all other zeros have to be concentrated in this one strip.

  • So, on one side of this strip is the vertical lineis the vertical axis.

  • And on the other side of this strip is this line, when real part is 1.

  • And so, let's... So this is called the critical strip.

  • These are all the complex numbers for which the real part is between 0 and 1.

  • And so, inside this critical strip, there's a middle line, for which the real value is 1/2.

  • And we look at all the numbers for which this is a real part.

  • For example, half plus five i will be a point somewhere here..., so it will be on this line.

  • And what Riemann suggested

  • the number of zeros is the minimum possible;

  • they all concentrate along this critical line.

  • According to Riemann's hypothesis, which still hasn't been proved.

  • His hypothesis is that all the zeros to his function lie on that line, except for these ones.

  • Exactly.

  • All the zeros lie on that line.

  • On that vertical line, which goes through the point 1/2.

  • That is exactly the statement of Riemann's hypothesis.

  • So one way to disprove the hypothesis would be to find a zero somewhere in that blue shaded area.

  • Exactly. We know that all of them are going to be in the blue shaded area.

  • The question is whether they are on this one specific line

  • And believe me, a lot of people have been looking for a counter example.

  • By the way, one wins a million dollars if one proves the Riemann's hypothesis, or disproves it.

  • So if somebody could find a point in herewhich is a 0 but is not on this linewill also win one million dollars.

  • So, a lot of people have been searching, but haven't been able to find.

  • Have a lot of zeros been found on the line? —Yes, a huge number.

  • I don't remember, but I think trillions of numbers.

  • So all the zeros that have been found are on this line.

  • And there's constantly a search going on for more and more.

  • The way I explained it, it sounds like an esoteric problem.

  • But actually, in this case, there is more to this, that meets the eye.

  • Because Riemann in that amazing paper that he wrote in 1859, he also explained that this behaviour

  • of his zeta function, which is now called after himRiemann Zeta function

  • and more specifically, the behaviour and location of the zeros of that function,

  • has a direct bearing on the distribution of prime numbers.

  • And prime numbers are incredibly important.

  • So prime numbers are, you know, something that people have been studying for millennia, right?

  • So Riemann was able to connect the properties of this function to the distribution of prime numbers.

  • And he obtained a beautiful formula, which tells you how many prime numbers there are,

  • say between 1 and 100, between 1 and 1000, 1 and 1 million,... any n — between 1 and n —

  • using his Zeta function.

  • Which is absolutely astonishing because if you think about the Zeta function...

  • something has to do with complex numbers, and with analytic continuation, and so on so forth...

  • So it's a particular branch of mathematics which is called complex analysis.

  • But prime numbers live in a different branch of mathematics; in number theory.

  • And it is a big surprise that actually the two things are very closely connected.

  • But this relation, that Riemann found, is predicated on the Riemann' hypothesis.

  • It's predicated on knowing all the zeros are located on this critical line.

  • That's why this Riemann hypothesis is so important.

  • Because it is only if we know that Riemann's hypothesis holds that we can obtain all those deep results about

  • the distribution of prime numbers.

[Prof Frenkel] Can I ask you a question Brady?

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Riemann Hypothesis - Numberphile

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    Prova に公開 2021 年 06 月 24 日
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