Riemann Hypothesis 3

This is part 3 of lecture on the Riemann Hypothesis. We're at the point where I explained that some infinite sums can be finite, although not all of them certainly are. And this one, 1 plus 1 4th plus 1 9th plus 1 16th, the reciprocals, the sum of the reciprocals of the squares of the positive integers, that was known to be a finite sum in the early 18th century, but nobody had a simple way to express what the answer would be. Like the geometric series summed to 2. Well, Euler made his name, his first claim to fame was solving this problem. And he said, amazingly, it's pi squared over 6, and this is pi in the sense of 3.14159. Stunning result. And he used some very, very creative calculus to prove this. He actually looked at it numerically and guessed the answer to figure it out first, and then figured out the calculus later. And this is when calculus was almost brand new. He also figured out if you take the 4th powers, 1 plus 1 over 2 to the 4th, plus 1 over 3 to the 4th, plus 1 over 4 to the 4th, the sum of all those guys, he has a similar formula, pi to the 4th over 90. Figured out a whole bunch of things like that. Now what does this have to do with our primes? Well, it turns out that you can look at the whole family of these answers. You take the sum, 1 plus 1 over 2 to whatever power you feel like today. So for example, the squares, you'd use 2. 1 plus 1 over 2 to the 2, 1 over 3 to the 2, 1 over 4 to the 2, etc. The 4th power series, the 4th power sum would be where s equals 4. Turns out zeta of 3 is still unknown. It's a very mysterious number. And it turns out you could even do this for s equals like a half. Turns out the half power of a number is just another name for the square root. It's a very, very flexible thing. And at this point, it seems like just a kind of a fun thing to think about, but not necessarily related to primes. Well, it turns out that it's deeply related to primes by what's called the Fundamental Theorem of Arithmetic. I'll abbreviate that as FTA. Fundamental Theorem of Arithmetic says, it's one of the most important reasons that primes are significant. 30, for example, factors into primes, 2 times 3 times 5. And that's the only way it factors, that it has the prime factor 2, 3, and 5, and that's the only way you could do that. 360, a bigger number, turns out to factor into 3 copies of 2, 2 times 2 times 2 or 2 cubed, times 2 copies of 3, 3 times 3 or 3 squared, times 5. You can do that if you want, multiply those together. 2 times 2 times 2 times 3 times 3 times 5 is 360. And this is the only way you can do that. You have to use 3 2's, 2 3's, and 1 5. And the Fundamental Theorem of Arithmetic says that any number has a unique factorization into primes. It says that primes aren't just things that you can't squish any further or take apart any further. They therefore become the building blocks of every other number. It's one of the reasons why primes are so important. So, let me show you what that has to do with this idea of the zeta function and summing things. What I'm going to do is I'm going to pretend for a second that the good old harmonic series, 1 plus a half plus a third plus like up to a 30th plus up to a 360th, all those numbers keep going forever. I know I said that was infinite. It is. But let's pretend for a second that it wasn't. What we could do is we could say that that can be realized as a product. So, this is where it gets a little bit hairy with the arithmetic. But bear with me here. What I'm going to do is I'm going to take geometric series. I'm going to look at 1 plus 1 over 2 plus 1 over 4 plus 1 over 8, only powers of 2. Add those all up. Now, that really is finite. Then I'm going to look at all the powers of 3, the next prime. All the reciprocals of those, add those up. Turns out, almost exactly the same proof as we had for the 1 plus 1 half plus 1 fourth. That's finite as well. And 1 plus 1 fifth plus 1 25th. That's the powers of 5 in the denominator. Keep going. So, all these are infinite sums. Okay, but they actually sum up to a finite number. So, that's nice. But that's pretty complicated. And then I'm going to take the product of these things in parentheses and I'm going to keep going for all the primes forever. Now, that's pretty weird. It's an infinite product. But to check that this has some relevance to us, we don't have to look at the infinite sums and the infinite products. I just have to look at the first few. I claim that this exactly is the same as just the sum of all the reciprocals. That's because, so for example, when you multiply sums together, is maybe a little algebra that would be helpful to remember. But when you multiply sums like this, you just multiply everything in parentheses here by everything in parentheses here. And everything here by, sorry, you multiply each thing here by something here, by something here, et cetera. So, for example, one of the things we're going to get is 1 times 1 times 1 times 1, et cetera, which is just 1. We're also going to get, for example, 1 half, choose the 1 half here, times 1 times 1 times 1 times 1, et cetera. That's going to give us a half. But we're also going to get, let's choose this one, 1 half times 1 third times 1 fifth times 1 times 1 times 1 times 1 times 1. That's a 30th. Or we could take 1 eighth times 1 ninth times 1 fifth. 8 times 9 times 5, that's 360. Because of the fundamental theorem of arithmetic, every number here can be built up as the product of a power of 2, a power of 3, a power of 5, if necessary. And if not necessary, you just use 1 for that prime, et cetera, for all the primes. And for every combination that I do here, I'm going to get exactly one of these numbers. And I'm never going to repeat. Because different combinations of primes give different numbers. That's the uniqueness of the fundamental theorem of arithmetic. Very cool. It's an algebraic version of expressing the fundamental theorem of arithmetic. That the sum of all the reciprocals, in a sense, I'm playing fast and loose with infinities here, is a product of geometric series based on primes. And of course, for the geometric series, it turns out that not only we can show these are finite, there's a very simple formula for the sums of all these individual guys. That's pretty cool. So we've got something to do with formulas, algebra, and primes. So that's promising. So here it is in official notation. Just wanted to show you how compact this can get if you do use the official notation. The sigma, this is Greek sigma, capital S. It says the infinite sum of the reciprocals of all the integers is, this is a capital pi coming in, again, pi coming in for product. And it's the infinite product over all primes of, and it turns out this is the compact way to get the formula for the geometric stuff. I just wanted to write that down because it looks so cool, but we're not going to work with that kind of algebra stuff. Now the problem is, of course, when you do this, this is infinite. And you know what? That might seem like a problem. It might seem like, hey, that means this is just bogus. I'm really not being rigorous, I'm pulling the wool over your eyes. But wait a minute. It turns out, let me go back, as I said, all of these guys are finite. Suppose for a second there were a finite number of primes. Remember, I never proved for you that there were actually an infinite number of primes. Most people believe it, but we should have a proof. Maybe there's some biggest prime, and then they stop. Every number beyond that is composite, a non-prime number. But suppose that this, and this, and this, we looked at all these things in parentheses, and there were a finite number of primes, and there would be a finite number of things to multiply here. And it's very easy to prove, absolutely rigorously, that this is a finite number, this is a finite number, this is a finite number, all these things in parentheses are finite. Well, a finite product of a finite number of numbers would give you a finite answer. But we also showed that this thing is infinite. And so that tells us, with just a little bit of tweaking to make it really, really rigorous, it says that this has to be an infinite product overall, an infinite number of primes, just like this is an infinite number here. Now, you want to make it more useful than that, not just saying, okay, we've proved this thing at Euclid nu. What you do is you go back to this zeta function idea. You say, what if I did this where I actually looked at the product, like the Basel problem, like one plus one over two squared, plus one over three squared, plus one over four squared, et cetera, or fourth powers, or third, or seventh powers, or half powers, or pi powers, or whatever. It turns out that there's a similar formula for that, that the zeta function, which seems to have nothing to do with primes, actually has a huge amount to do with primes. It turns out to be expressible as a product of primes. Okay, so I'm gonna have to leave that thread for a second. But what I want you to get out of that is that there's this thing called the zeta function, and it is relatively simple to define, and it has a deep, deep connection with primes. So, another seemingly totally unrelated problem. Of course, the punchline is gonna be that these all are related. Fourier, not a mathematician per se, really, more of a physicist engineer. He was looking at the problem of analyzing the flow of heat. It was a huge open problem then to have a mathematical description for the flow of heat, and actually get some answers out of it. People had thought about what it should be, but people could not get very sensible answers out of it. He had a beautiful insight into it, and it was so striking and so unexpected that people thought for 20 years or so that it really was completely ludicrous, and was meaningless, and was hocus-pocus, and wasn't gonna amount to anything. They were completely wrong, and he eventually convinced them. What he did was he looked at functions. He looked at the graph of a function. If you look at the GNP of a country as a function of time or something, anything that associates one quantity to another. Here's time, here's GNP, anything. Position, maybe, again, as a function of time of some object that's going up and down, and then stays down, and then goes up suddenly, and goes down. Functions are the bread and butter of modern mathematics, and you can also look at, for example, the temperature along a rod. Suppose this rod had, part of it was in an ice bath, and then part of it was really hot, and then the other part was in super dry ice or something. All kinds of different kinds of functions you could get, graphs you could get. Let me pull that up a little bit. So this is like the temperature as a function of the position along the rod. Well, what Fourier said is, functions can be very complicated, but I can use sine waves. Simple, famous kind of wiggle, this kind of wiggle, the red here, as building blocks for functions. And he said that any curve, any possible function graph like this can be broken down into a sum of certain special kinds of wiggles, sine wave wiggles, a very special kind of pattern. But here's something that's interesting, that these sine waves are very smooth. They have no leaps, they have no jumps up and down. That's very different from this white curve, which is called a square wave. This has discontinuous jumps, it suddenly jumps up and down. If you think of it as like something that varies in time, it's like something that goes from low instantaneously to high, instantaneously back to low and back and forth. It doesn't seem like if you added together these kinds of wiggles, even if you added a few of them together, like maybe I've got a big wiggle and then I've superimposed some little wiggles on it, it doesn't seem like you could actually ever get something like the white curve. But look what happens if I keep taking more and more curves, more and more wiggles. What this is doing is it's taking that original big wiggle and adding small amounts of tinier and tinier wiggles. And what happens is that amazingly, it becomes very, very much like that white curve, even though the white thing had these sudden jumps. So that's pretty interesting. And for some functions, it's actually surprisingly easy. This guy doesn't have any sudden jumps, but it has sharp corners. But if I first try to approximate it with this one red wiggle, a single sine wave, and then I add in a little bit of extra wiggliness, you can hardly even tell, but it's just a little bit wiggly on a little bit of a finer scale. Then I add just a couple more. You can't even hardly tell the difference anymore. That, what the red curve here, is all built out of very smooth wiggles, and yet it's got, approximating something, it's got sharp corners. That's the basic Fourier idea, is to take any kind of pattern, any kind of function, basically any kind of graph that you can imagine, and make it out of a simple combination of incredibly nicely behaved smooth pieces. Basically, Fourier can do anything it wants to do. Even these kinds of jumpy functions, if we go back to this square wave idea, it can do almost anything. It's really, the further you press it, it's amazing how far this idea can be pressed. So this is the idea, it's called Fourier series. A series is like what I was talking about with harmonic and geometric series, an infinite sum. The idea is that mathematicians like to say, well, I'm not gonna just take a finite number of these things, I'm just gonna look at the infinite sum. And the idea is that you basically recover exactly the white out of an infinite sum of these wiggles that give the red curve. Turns out there's an even more flexible version of that called the Fourier transform. And you probably just wanna think of it, I don't wanna make the distinction very explicit, just think of it as the idea of using standard, simple wiggles to create an arbitrary picture. And here's a totally wacky idea. I'm not, I'm just gonna bring this out of nowhere, even though it's really, yeah, I think this is the fundamental idea of this whole talk. Could we find, what if we had one of these really wacky functions, where was it? Remember these functions like pi of x or psi of x? Let me give you the psi of x function. Remember this Chebyshev function, which was definitely a sharp cornered wiggly function, discontinuous, it jumps, it's defined to jump at all the primes and prime powers. Could I actually find, what if I did the Fourier idea here and tried to use standardized smooth wiggles to approximate this function? What would happen? Would it give me any insight into the actual jumps in the pattern here? That's a very interesting idea. And would it at least show me that it has this behavior where it's supposed to be going along a straight line? Hmm, that'd be useful. Okay, so let's, that's a good place to stop this segment of the talk though.