Riemann Hypothesis 1

All righty, I want to talk about the Riemann Hypothesis. This is a talk that I gave to a senior group called Oasis. And the Riemann Hypothesis is one of the $7 million Clay Math Institute Millennium Prize Problems. It's the most famous of them. And many people would call it the most famous unsolved problem in all of mathematics. And I am not an expert in the Riemann Hypothesis. And I'm purposely giving a very stripped down kind of non-technical discussion of this. But I hope I can convey the thing that I think is cool about the ideas here is that with almost all cool mathematical ideas, the interaction of very different and unexpected pieces of mathematics and the unexpected way they fit together. And the other thing about the Riemann Hypothesis is that even though we'll see that it gets definitely goes off in kind of an interesting direction, it's really about a very, very basic issue. The first question is we want to understand the prime numbers. Those are the numbers that have no factors besides themselves in 1. They're the building blocks of the integers and of arithmetic and multiplication and division. And here's a list of the first few. And then here's a couple, 257 to 63. It's easy enough to figure these out by hand at the start. I think I've figured out with my little son, we figured out the first few primes. And he definitely got the idea. But what about very large primes? How does this sequence continue? So for example, how many are there total? Are there 100,000? Are there an infinite number? That would be interesting. If not, it'd be even more interesting probably. What is the 1 millionth prime, for example? That's not an easy question to do without a computer. How many primes are there under a million? These would have been very, very tough questions before the age of computers. Well, because we have computers, we can answer some of those numerical questions. The first thing to address is most people would guess, and it's true, that the primes never stop. There's an infinite number of them. That goes back to Euclid over 2,000 years ago. And we'll see a proof of that as we go. It's a cool proof that comes out of the ideas that we're going to discuss. What's the millionth prime? 15 and a half million, about. And there's about 78 and a half thousand primes under 1 million. And what we'd really like to do is not just have random, just factoids like that, but look at the patterns, and especially the patterns when these numbers get really big. Now, I'm going to change it just slightly, though. I'm not, instead of counting the total number of primes, like this 78 and a half thousand, under a million, turns out that it's going to be easier if we look at a moving window. So what we're going to do is we're going to look at primes near some big number. So, for example, here are three primes just a little bit under 10 billion. Here, they're only two apart from each other. It's called a twin prime pair. You can't have primes, consecutive numbers, because then one would be even, and only two and three. That's the only time that happens. But you can have them two apart from each other sometimes. It's called a twin prime pair. They're very interesting. And then here, the gap, the next prime turns out to be 46 up. That's a much bigger gap. Interesting. So here's what we're going to do. We're going to take a moving window. So look down here. We're going to look at a number line, and this isn't going to be true to scale exactly, but maybe from zero to like a billion. And we're going to look at a moving window of fixed size. So here's our window, just that much. And let's say that's 1,000 units long. So I could go like here, zero to 1,000. Oh, that didn't work, did it? Sorry, didn't see that. There we go. Here's just the window length. And so I could do that length from zero to 1,000, or I could locate the window at a million. You can tell it's not true scale, I know. And just do 1,000 in here. So that would go up to 1,000,000 plus 1,000. Or I could put it at 10,000,000, same size window. And so this guy would be 10,000,000 and 1,000. And so we're always going to use this window just for convenience and move that window to different positions in the number line higher and higher and see how many primes are there in here. How many primes are there in here? How many in here? How many in say the same size window up here? Okay. So for example, how many primes are there between a million and a million, 1,000, a billion and a billion and 1,000? Or one followed by 100 zeros. That's what this notation means, 10 to the 100th power. One followed by 100 zeros and that plus 1,000. Another way to say it, and this is very closely related, if I knew how many primes there were in between here, say we'll find the number in a second, but say there were 20 primes in between a million and a million, 1,000. Then the gap, there's 1,000 numbers, sort of 1,000 units along the number line down here. And if there's 20 in there, then there'd be a gap of about 50 between each one. And so they're very closely related, how many primes in a certain window to the average gap size. And that's what we're going to actually really going to be interested in in a minute. Is there a nice formula for the gap size? We don't expect a precise exact formula that will give us the gaps, like here 2 and then 46 suddenly. It's just too complicated. But is there some nice formula on average? And then if we can get that, what about fluctuations from that average? Like if you have some sort of random phenomenon, maybe like winning the lottery or something, you can predict on average how many times you're going to win. But all the individual winnings, that's harder to predict. And that's the fluctuations from the average. This is really the whole thing in a nutshell. The known stuff is mostly about the nice formulas for averages and then what's still unclear and where the Riemann Hypothesis lies. It's fluctuations from that average. And okay, so here's the answers. It turns out there's actually 75 primes between a million and a million, 1,000. There's 49 between a billion and a billion, 1,000. So here the gap size is about 20, 1,000 divided by 49. It's about 1,000 divided by 50, so it's about 20. And then here, that's a hard one. Exact numbers for this? I don't know if anybody knows that. It's just way, way big. So as I said, what we're going to do is we're going to look, what we'd like to look at, not just the number of primes in here, but take 1,000 and divide that by a number of primes. That gives you an inverse measure of how dense the primes are, how spaced out are they. So let's do some more systematic stuff, okay? Let's look at a table. We're going to have N. Let's just pick really simple numbers, all powers of 10, or 1 followed by a bunch of 0's, 3 0's, 4 0's, 5 0's, 6 0's, 7 0's. We'll use our window of size 1,000. And then we'll count the primes in that window. Now, notice, of course, this is something you'd have to do a lot of work to do or have a computer do a lot of work to do. But let's assume that work's been done and it's been done correctly. You look for a pattern here. Well, you can pause for a second. The only pattern I would say is obvious here is it's just going, it's decreasing, but we'd like to know more than that. So primes are getting more sparse. That makes sense because it's harder for a big number to be prime because it could be divisible by all kinds of anything below it. Then what about if we take the numbers a bit bigger and we do what I was suggesting before, which is instead of just counting how many primes, look at the gap size. Now, you might want to pause again and see if you can see some sort of pattern here. Especially, look at what's happening here. These numbers are growing very, very large. Every time I go to the next row, I'm multiplying by 10. And here, these numbers are not growing in a multiplicative way. It's not like 7 or it's not like, you know, 10, 20, 40, 80, it's 160 or something like that. It's more in an additive way. Looks like we're going up by 2 or 3 each time, roughly. Well, let's think about that. There's something here that's just going up by 1 each time. Can you see what's going up by 1 from here to here to here to here to here to here? It's the number of zeros. Aha, so that's what we're going to compare. That's what we're going to do. We're going to look at this number, the gap size. Really, it's 1,000 divided by the number of primes, but it's basically the gap size. And we're going to compare that to the number of zeros. A good way to compare it is just divide. And look at these numbers. They're all very similar to each other. And in fact, these numbers, especially down here, they're getting very close to 2.3. There's still some fluctuation, of course, but there's definitely a pattern. There's some sort of magic number 2.3 that's showing up here. Now, one thing we could do is we could change the window size finally. I've always used a window size of 1,000 so far. Let's change the window size to a million, so a million to 2 million, or a billion to a billion plus a million, or a trillion to a trillion plus a million. We're going to get a lot more primes in a bigger window. But if we look at the gap size, compare it to the number of zeros, divide, we get numbers that are really very close to 2.3, especially these last two, very close to 2.3. In fact, the magic number turns out to be 2.303. Gauss, Carl Friedrich Gauss, best mathematician who ever lived, or at least one of the very, very best. In 1796, predicted, he didn't publish it for years, but he predicted that the spacing of the primes on average is roughly 2.303 times the number of zeros in the number that tells you where you're doing your window. And so, the primes around a trillion are spaced at about, well, a trillion is 12. You just take 2.3 times 12, the spacing between the primes is about 2.3 times 12. And I'll explain where this magic number 2.303 comes from, or somewhat I'll explain it. And the prediction is that this isn't a very good approximation for small numbers. But as n gets larger and larger, it's a better and better approximation. So, let me say a little bit more about that. This is a little bit more technical, and I'll go through it sort of quickly. But it just brings up a great issue that I always talk about with my students. Remember, I've been counting in a window. You can also look at the actual running total, all the primes less than x. And that's how this statement is usually stated. If you look at the running total of all the primes, that's what this is. The number of primes, this is a set, all the primes less than x. So, like pi of 1,000, this pi is an unfortunate notation because we are going to use pi later as well in the sense of circles and geometry and things like that, although it's going to come in in a weird way. Let's look down here. So, for example, pi of 1,000 is just the number of how many primes are there less than 1,000. It's basically how we started thinking about it, and then I switched to the window idea to talk it to bring up the idea of gaps. But it turns out, it's good to use this too, that the running total of primes. It turns out that there's this function li of x. And basically, that's what the running total would be if the primes had exactly this behavior. If the primes really exactly had this behavior that the gaps was 2.303 times the number of zeros. And then you figure out what does that mean for the running total. It's not too hard to do, to translate that. Turns out, it's called the function li. And Gauss predicted, what he really said was the running total of primes in reality is close to this very simple function that basically says the gap has to do with the number of zeros. The exact meaning of this approximation is that if I take the ratio of the prime counting function and this fairly simple li function, the ratio goes to 1 as x goes to infinity. Important that that is percentage error. It's a relative error. He didn't say, these two numbers actually got smaller in the sense that their difference was going to zero. That would be an incredibly precise statement. So let me talk a little bit about the difference between relative and absolute error. Suppose I have, well, let's say Al has a billion dollars. He's a billionaire. And then somebody gives him a gift of $100,000. So now he has a billion and $100,000. Is he jumping for joy? Does that matter to him much? Not much. Because relative to the billion dollars he already had, it's not that much difference. And one way for him to calculate that is he's going to say, by what ratio did my wealth increase? If you do that ratio, it's just 1.0001. That's very close to the number 1, which would mean it wouldn't change at all. That's a relative change. Or another way to say it, if you're talking about error is suppose somebody, suppose his accountant thinks he has exactly a billion dollars, but he really has a billion and $100,000. Is he going to fire the accountant? Probably not. Because he didn't make that much of an error in a relative sense. What about Bob? Bob, let's say, has, starts out with $100,000 and then gets a gift of $100,000. Or another way to say it would be suppose his accountant thinks he has $100,000, but he really has $200,000. In a relative sense, that's huge. He has twice as much money as his accountant thinks he does. He should probably fire that accountant. Okay. So relative error, even though you added $100,000 in both cases, this is a much bigger error. If I look at the absolute change in both cases, I subtract to get that. And so, for example, if I actually say, okay, how much money is the difference? Like how much money did Al actually receive in this gift? If you look at the scenario where it's a gift, it's $100,000. Same here. The difference would be $100,000. So when you measure absolute error or absolute changes by differences, and that's a very different mechanism from looking at relative error. And most of the time, with really huge numbers, like billions, you're more interested in relative error. Yeah, okay. I was off by $100,000. Wow, that sounds like a lot. Until you realize we're talking about super big numbers. So it turns out that Gauss said,