[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 out — and it’s very easy to show — that 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 which — which 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 okay — can 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 of — what we call holomorphic — functions 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 allow — which enable us —
to 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 be — you 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 number — uniquely 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 calculate — will 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’ heel — that’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 about — for what values — for which s we have zeta(s)=0. That is the one million dollar question. For which value of s do we have — does 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 reason — it 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 else — where else are the zeros — where 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 line — is 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 here — which is a 0 but is not on this line — will 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 him — Riemann 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.