Riemann Hypothesis 5

Part 5 of the Riemann Hypothesis. I want to go back to one thing I said at the end of the last part. I want to just show you a couple pictures of the idea of what with a kind of map of the possible growth and oscillation of f in the complex plane. What would that mean? The map and the special points on the map. Well, let me show you this is actually a certain kind of picture of the function, the map, that we're actually going to end up getting and I just wanted to give you just a very quick thing to have in your head visually. I'm not going to explain all the colors and everything, but you can see that it's a plane. It's a coloring of the plane, and there's different colors and also different intensities. Some are dark, some are much lighter, and that's giving a lot of information about what's going on with a certain function in the complex plane. And I just want to point out, there's a couple of, there's a few special points where the colors all come together and there's a little black point right in the middle. There's also a very dark portion in here and there's a bunch of interesting stuff where colors come together here. You can kind of see some black points there. It's going to turn out that these particular black points are going to be really the crucial thing about this particular map. Again, it just turns out that when you apply, remember the Fourier and Laplace idea, when you combine those using complex numbers, you can get a picture and basically what these dots, these black dots, are the most important thing definitely about this. They give you a map of what kinds of oscillation you could find. It also has the growth properties encoded as well, but what's more interesting, once you get past the basics, is the oscillations. Remember that we have this picture. We want to show that, for example, the prime counting function, first of all, has this simple dotted line behavior as an average. That's the prime number theorem. That's certainly a big deal and you use this kind of technology to prove that first. But nowadays the modern interest, once you prove that, is how do you quantify the oscillations around that average? That's more of the oscillatory behavior and that's more encoded, it turns out, by these dots here. Another picture of this same thing is you can think of this as a landscape. The complex plane is kind of the bottom of this picture. We're thinking of taking parts of that plane and raising them up high and lowering them. We're still coloring them, which is interesting. Here you can see the colors coming together at a little pit. That's exactly one of those black dots where the colors came together in the other picture. This gives you a little bit of a different picture. You can kind of turn it around. This is a small section of the thing and I can show you a bigger section by a zooming out process. You can see that there's a bunch of waves and wiggles in this map. The connection between the waves and wiggles in this map and the wiggles in the prime counting function is definitely a tricky one. There's a lot of stuff involved there, a lot of real math there. But there is a way to encode the connection between those two. Those are kind of cool pictures to look at. What is that map? I've kind of given away a little bit of a punchline that we actually can graph this thing. We can actually go ahead and find out something about this map. Right now, we seem to have this really complicated way to get there. We start with the prime counting function. We apply the Fourier Laplace thing. That's pretty complicated. We get a map. How do we know whether we can actually calculate that? Maybe it's just going to be totally mysterious, of course, because remember we don't really have a good formula for the prime counter. Either pi, the normal prime counter, or psi, the Chebyshev modification one, which looked like it fell on a straight line more or less. If we don't have a formula for that, how can we even start and do the Fourier Laplace? If you look at the start of the story, like you open up a college textbook about Laplace transforms or Fourier transforms, they're going to assume you have a formula for the thing you're starting with. If you don't even have a formula for the thing you're starting with, how can you even start this process? And if you had a formula, we wouldn't have to do this. So, that's a big question, but that's where what's called the zeta function comes in. We introduced it a little bit while ago, and it turns out there's a beautiful connection between the Laplace idea, the Laplace transform idea, and the zeta function. The definition of zeta was I pick whatever favorite exponent I like, and then I take the sum of the reciprocals of all the positive integers raised to that power. When s was 1, we got infinity, but it was an interesting kind of infinity because it was the harmonic series. It's the simplest kind of series of decreasing terms when you add them up to get infinity. When s is 2, it's the Basel problem, which Euler got an answer for. Well, now we're being much more ambitious. We'd like to understand this for any s, and in fact, we're going to think about what would happen even if s is a complex number, whatever the heck that might be. Turns out that this is a special example of a very general idea of taking any sequence of numbers you like, like 1, 1, 1, 1, 1, or 1, 2, 3, 4, 5, 6, 7, or here's a weird sequence, 1, minus 1, minus 1, 0, minus 1, 1. I'm not going to tell you the pattern for that, but it turns out there is a pattern, and it's a cool one. See if you can guess it just from the first few. It's pretty hard. You can pick any sequence you like of numerators and put them over these special denominators and add them all up. Usually, it'll make sense, at least for most s's. Those are called Dirichlet series. What it really is, is suppose I have an interesting sequence of numbers. Usually, they come out of interesting number theory problems, like this bizarre series of 1's, 0's, and minus 1's. What I'm doing is I'm basically comparing them to the possible denominators I could get, because remember, if I let s be anything, this could be 1, minus 1, minus 1, 0, et cetera, over 1, 2, 3, 4, 5, 6, et cetera, or 1, 4, 9, 16, 25, et cetera, or 1, 8, 27, 64, et cetera. The different answers for the sum that I get for the different s's basically compares the sequence on the numerators to those different possible, very standardized denominator sequences. I'm going to compare some interesting sequence of numerators that I don't understand very well to these standardized sequences, the denominators with s equals 1, s equals 2, s equals 3. What's going on here is that I'm comparing a standard sequence. It's like a function, it's just I only have sort of a discrete set of numbers. I'm comparing that to different rates of growth, linear growth, quadratic growth, cubic growth. This is basically the Laplace idea. It's just a discrete version of the Laplace idea. I have to say that the way you relate, there's two tools you'd have to use to relate that. This kind of growth, these kinds of growth are not exponential growth. They're like raising functions to some power. That's a different thing, but you can relate them using, guess what, logarithms. Very cool and essential tools we've seen already. And a little bit of calculus, not a huge amount really. Just, you know, it'd be a hard Calc 1 problem or Calc 2 problem, but it's nothing really, really deep to show that this is not just essentially Laplace. You can translate it into the Laplace idea. And so the zeta function turns out to encode a discrete version, essentially, a number theoretic version of the Laplace transform. Okay, so here's some links. Again, just on a very formal level. I can't show you the details of anything. We start with the primes. We use the pi counting function, which is just the running total of the primes. We've changed that to the psi counting function because it's a better behaved counter. Turns out if you take the Laplace transform of that psi counting function, you will get the zeta function. And that's what I wanted to kind of suggest might be true based on the fact that in the previous slide I claimed there was this affinity, a very strong affinity between the Laplace idea and the zeta function. So it turns out when you do the Laplace of this very special psi function, that's a whole sequence of numbers. And if you do the Laplace idea, it's essentially recreating the zeta function itself. Okay, but what other links do we have? If somebody had said, hey, take the Fourier transform of the psi function, that would give me information about psi, especially about how it wiggles. But, oh, this stuff up here said, no, we're not taking the Fourier transform of the psi. We're not looking at wiggles in psi. We're looking at the growth rate of psi. And the lucky, lucky thing is we got the zeta function, which has a very simple definition. Well, I also told you that Fourier, analyzing wiggles, can be related very directly to analyzing growth. That's the Laplace idea. You just put in complex numbers. So the surprising idea is that, hey, this is kind of a cool thing to do for s equals 1, 2, 3. I add up different kinds of reciprocals of numbers. It's kind of fun to play with. Could I possibly make si or 3 plus i or 17 plus pi i? Well, it does actually make sense. And you can compute with it. And it turns out to be the complex zeta function. So if you understand zeta well, you will understand the primes well. That is the whole punch line of this whole thing. And that's where I've been trying to get to. If you understand the zeta function really well, you understand the primes well. And you have to understand it for complex values of psi. Because if you don't use that, you're never going to see the connection between Laplace, the growth analysis, and Fourier, the wiggle analysis. The right-hand side, what do I mean by that really? It really means what happens for large primes. It's not going to give you stuff for medium-sized primes. It's not going to give you the absolute details. It's not going to just tell you exactly where every prime is, unless you understand zeta unbelievably well. If you did, then it would actually give you that too. But if you're focusing on questions about how many large primes are there, how do they grow, some other patterns. There are deeper other patterns that you could theoretically get if you understood zeta really well. And that's where Riemann comes in. He is the one who realized that you wanted to take the zeta function, which Euler had come up with, and people already knew was a good way to think about primes. He said, let's use the new theory, which I helped invent, of complex analysis, using complex numbers and calculus, to take zeta into the complex plane. Let S, the argument of zeta, be complex. And then he showed how that would give you an incredibly good understanding of the primes. And so he basically sketched out how you should prove the prime number theorem. Remember, that's the fairly simple statement that, for example, this Chebyshev pi counter, the psi function, should be on average a straight line. That in itself is a profound achievement of the 19th century. He didn't quite prove it, but he laid out how people should prove it. It took a while to get the details down, to really do that complex analysis, that complex number calculus well enough. But it was basically his idea that they followed up. And he also made this big conjecture that would say, I think that the wiggles are pretty predictable. And specifically, it was a prediction about where the little dips are on this picture, or equivalently, where these black dots are. Notice they line up on a line. It turns out they line up on a line that's just a little bit to the right of the axis that contains the pure imaginary numbers. There's a zero here. You might not be able to see it on the YouTube. But the pure imaginary numbers, just like 1i, 2i, 3i, minus 2i, minus 5i, lie right in here. If you just go a tiny bit to the right, you see that's where these dots are. Those are the zeros of the Riemann zeta function. They're the non-trivial zeros. There's a few others that don't count. And it turns out that the fact that they're a little bit to the right exactly encodes the growth, because the real axis here, like real numbers go left to right, that's where Laplace comes in. You go left to right, you're talking about less growth, more growth. You go a little bit to the right, there's a moderate growth in the pi and the psi counting functions. It doesn't blow up. It's more like linear growth. And then the exact where the dots are gives you something about the wiggles. And Riemann said, just like it looks like from this picture, those dots really, really are all lined up on a straight line. And that would say that there's really no unexpected weird properties of the distribution of primes. And here's a picture of the man himself, Bernhard Riemann. And he's very famous for many other things, including, for example, the kind of geometry that leads to general relativity. One of the best mathematicians ever. Died young, unfortunately. So, Riemann said the map, this map in the complex plane, this picture that we're looking at here, it clearly shows no surprising growth in psi. He made clear how to show that the psi counting function is approximately just equal to x, very simple straight line behavior. He said, if you can show that these dots don't get a little further over to a dangerous place, like right around here, then you're done. He also conjectured that these guys, this is the real part of s, he said, they actually have real part one half, exactly on the line shown in the picture. And the Riemann hypothesis says that all the non-trivial zeros of psi, so some zeros except for the ones that are totally well understood, have real part one half. 30 years of slow digestion followed from that. 1896, Hadamard and de la Vallée-Poussin proved the prime number theorem, which was using this, that they don't go over to position one. Didn't have to use this much stronger thing, that in fact, they don't even get near one, they actually are all on the line of location one half. The Riemann hypothesis would imply not just that psi of x is kind of close to x, that the simple prediction for primes, remember this corresponds to the 2.3 times, counting the zeros kind of thing. You would say that it's really close. It's a surprisingly good approximation. It would also show that, of course, there's going to be deviations from that. The primes have some wiggling to them. It would show that besides the patterns we've already found, they're basically randomly distributed. They're definitely not random in an absolute sense. No even number can be prime except two, for example. But besides the patterns we already understand well, Riemann hypothesis would say that everything else is basically pure randomness. It would make many, many other conjectures true about the primes, and it would make a lot of analysis easier of the primes. That's a big reason for its importance. Primes nowadays, and a lot of other number theory, used to be the purest of pure mathematics. It wasn't useful for anything. But nowadays, it's incredibly useful for cryptography, which is really, really important, for internet commerce, for national security, for business security. There's not any direct connection anyone knows necessarily that says if someone proved the Riemann hypothesis, if the National Security Agency knew a proof and nobody else did, there's no obvious way how they could turn that into a cryptographic bonanza to either make codes or break codes. But people would definitely have to pay attention, and you never know. Maybe a few years after, or maybe the ideas in the proof of the Riemann hypothesis would either save or kill cryptography. I mean, it could be a very big deal. It's very hard to be sure. A little bit of the history of this to close out. In 1900, Hilbert, David Hilbert, one of the best mathematicians also, made a list, a famous list of open problems, and Riemann hypothesis was on that list. And from the next 30 years, lots of really good progress on it, especially by Hardy and Littlewood, two British mathematicians. And they came up with lots of precise results, lots of conjectures, but certainly no real... They weren't close to the proof of Riemann, definitely. 1949, Selberg came up with a totally different proof of the prime number theorem from the one about 50 years ago using Riemann, using complex analysis. And so it turns out you don't absolutely have to use this kind of complex variable idea, at least to prove the prime number theorem. But I think very few people would say that you don't want to use it to go further than that. 1920 to 1970, I can just make a bare mention, huge, huge, huge generalizations of number theory and algebra, and especially what's called algebraic geometry. Huge, massive analogs of Riemann hypothesis. And most of those are now proved, and it's kind of ironic. Those are much more complicated to understand and state, and they have great consequences, but in some sense they're significantly easier to prove. And that's one of the reasons why people believe that Riemann should be true. So many similar results have been proved rigorously. It's not the best reason to think of it, but... In about the last 50 years, there's some wonderful connections to quantum mechanics and especially things like quantum chaos that have been tantalizing, haven't resulted in a proof, but really beautiful connections to physics that are totally unexpected. All the last 110 years, people have tried to do numerical evidence. There should be a C in there. They've done by hand, and then they've done by computer. Remember, the precise statement of the Riemann hypothesis is that when you look at this pattern, and you look at... Let me see if I can zoom in here. Nope, it doesn't like that at all. Okay, I'll zoom in a little bit. These zeros are supposed to line up exactly on a certain line, where all the numbers you could build out of imaginaries and reals, but the real part has to be exactly one half. And that's verified up to the first 10 trillion zeros. A guy named Odilisco has done that. He's the master of computing with the zeta function. The question is, well, isn't that enough? Can't we just stop and say, gosh, it must be true? Well, it turns out that this field of mathematics is exactly a good place to look for why you shouldn't be so confident about that kind of thing. A few people, Littlewood, who I mentioned before, a guy named Skews and some others, they found that there's similar conjectures. There's conjectures about the behavior of this function. It's not the Riemann hypothesis, but other conjectures you could make, and people did make and said, oh, I'm pretty sure this is true. They start to fail only when you look at numbers like one with 316 zeros after it, 10 to the 316th power. That's an unbelievably large number. Odilisco is nowhere near that. So you could say, oh, we think it's true because we've got evidence of that. Well, you could have said that about the Skews, about the Littlewood conjecture, and Skews came along and said, oh, no, it looks really good, but I can show you that there's going to be a counterexample if you look at unbelievably huge numbers. In terms of Riemann, there's suggestions that maybe the first place a problem would occur, where you wouldn't get a zero on the critical line, where a real part is one half, it might be 10 to the 10 to the 10,000th. So take one followed by 10,000 zeros. That's an unbelievably huge number already. And then take one followed by that many zeros. We just can't access that computationally as far as we can tell, unless we get immensely cleverer. And so we're going to have to have some better ideas. And really, who knows if it's true or not? There's smart people out there who think it's false, even though the general opinion is that it's probably true. Hopefully, in the next, you know, 20, 50, 100 years, someone will prove it. And that's the end.