Riemann Hypothesis

Here's part two of the Riemann Hypothesis. We're getting well into the 19th century here, in terms of the history of this, where Gauss, right before the 19th century and then through the 19th century, people tried to prove this statement that a relatively simple function based on counting zeros can predict, on average, roughly how many primes there are, and things like what's the spacing, what's the gap between the primes. But in the sense of relative error, not in the sense of absolute error. So now, I want to say more about this zero counting function. It's a very, very basic, important function in all of mathematics, and it doesn't usually go by the name of zero counting. Basically, because it's well and good to say for 10 that has one zero, for 100 has two zeros, for 1000 has three zeros, but what about any numbers in between? Like 987, that doesn't have any zeros in it, but should probably give you something close to three, because 987 is close to a thousand. Can we fill in more of this table? Can we fill in something in between a hundred or a thousand or between ten and a hundred? Well, I want to notice something about these numbers, that when we multiply ten and a hundred, we get a thousand. And what happens to the number of zeros? One, not times two, but one plus two is equal to three. So, the multiplication over here corresponds to addition over here. So, let me write that down. And so, that actually gives us a hint as to how to fill things in. For example, suppose we could find a number that multiplied by itself gives you a thousand. Then, that should correspond over here to something when you add it to itself gives you three, or in other words, one and a half. So, to fill in right in the middle here, with a one and a half here, all we have to do is find a number that multiplied by itself gives a thousand. In other words, the square root of a thousand. So, 31.6 times 31.6, do it on a calculator, you'll discover it's about right close to a thousand. And I claim that that should be the thing that corresponds to 1.5. So now, again, this zero counter function idea, that name becomes not very accurate here. But it's an extension of that simple idea of counting the zeros. This is the number that, in a certain sense, is halfway between ten and a hundred. It's halfway in that, another way to say it, let me do this on the paper here, is that if I look at, I start with ten, and I multiply it by a certain magic number, 3.16, I get up to 31.6. Why is that such a magic number? It's because when I take 31.6 and I multiply it by that same factor, again, check it on a calculator, I get all the way to a hundred. So, multiplying by 3.16 twice gave me, in one step got me to 31.6, and another operation of the same exact type got me to a hundred. So, in terms of multiplicative steps, not kind of adding repeatedly that we're much more used to, like counting by twos or counting by fives or counting by tens. If instead you always multiply by the same number, then what's a halfway from ten to a hundred? It's 31.6. And what's the step size? It's 3.16, which is the square root of ten, by the way. So, there's a way to extend this to get meaningful numbers for everything. Not just 31.6, not just ten, a hundred. There's a way to take any number and sort of say how many zeros does it have in a generalized way. And this has an official name. It's called the logarithm, or the logarithm to the base ten, because I've been using ten as our fundamental system here. That's not embedded in nature. That's really because we have ten fingers. But that's one way to do it. The log of 10,000, of 1,000 say, is you write 1,000 in the usual way, using powers of ten. It's one followed by three zeros. Or another way to say it is it's ten to the third power. And then you just say, okay, if it's ten to the third power, I'm going to say the logarithm is three. The log of 31.6 was not nearly as obvious. You couldn't say it's zero counting, literally, but it turns out to be 1.5. Now, I promised you an explanation of the 2.303. So, very briefly, it turns out that there's a better base to use. There's a better system of counting to use than ten. It's not even an integer, as it turns out. It's not even a whole number. It's e, which stands for Euler. It's 2.718, et cetera, et cetera. It's this very strange number. It's irrational. It has a never-ending, non-repeating decimal expansion. But it turns out that that has much better properties. And we might see a hint of those better properties as we go. And what you, so what you do is you use powers of e instead of powers of ten, like 1,000, to express things. I'm not going to say any more about the details. But the key thing for us is that it's not really that different from counting zeros. To translate between them, you don't have to do something completely different to calculate this natural log. Ln is the abbreviation for natural log. It's just 2.303 times the ten, the zero counting function. And so, for example, 1,000, you just take this three and multiply it by 2.3. You get about 6.9. And so, that's where the 2.303 came from. And this is exactly what we were looking at in the table. The spacing between primes looked like you take the number of zeros in the number where you're looking, multiply it by this weird magic number. That turns out to be exactly the natural log. And so, what's called the prime number theorem basically says that the spacing between primes for large, when you look at large, large numbers is about this magic function, the natural log of x. This, the natural log appears every, every, everywhere in mathematics. It's incredibly important. And this is one of the many places it comes in. Okay. So, yeah, logarithms actually are all over this subject in particular. And so, I wanted to highlight one more variation on how to count primes. We've seen two already. We've seen, let me go back a sec. We've seen the pi of x is the running total of all primes. So, pi of 1,000 is all primes up to 1,000. We've also seen things like how many primes are there in a window of a fixed size. Well, once you discover that logarithms are all over this business, it turns out, and I'm not going to really explain why. It turns out that Chebyshev, Russian mathematician from 19th century, he's got another one called psi of x. This is the Greek letter psi. He had a slight variation that's just going to make for prettier pictures. Pi of x, what it does is if you imagine sort of making a table of pi of x, it doesn't, every time you increase x by 1, pi doesn't necessarily jump at all. But it jumps by 1 every time you have a new prime on the list. Psi of x, it turns out, it's smarter to jump by the natural logarithm of the prime that you just discovered at each prime and not jump at all. Except it turns out that if you have like prime powers like 3 squared or 5 to the 4th, it actually jumps there too. So, let me show you the graphs of those guys and explain a little bit why you would want to count in that way. So, here's the pi function. So, what happens is at 2, it jumps up 1 and then at 3, it jumps up another and then at 5, it jumps up another and then at 7, it jumps up and then 11, it jumps up, at 13, it jumps up. And so, at every point like pi of 15 is 6 which says that the number of primes less than 15 is 6. It's just the 6 primes, 2, 3, 5, 7, 11 and 13. And it turns out that if you start looking at this in a larger and larger scale and the applet works, there we go. My computer is doing a lot right now with the recording on. It turns out that that looks like a fairly nice simple function that kind of curves in a gentle way. Turns out that the idea that it's basically approximating that dashed line is the prime number there. But that line is a little bit complicated to talk about what the graph is. Turns out, if instead of doing the pi function, we do a little bit of a weirder thing. We do the psi function. So, here at 2, it jumps a little, at 3, it jumps a little, at 4, because that's a prime power, it's 2 squared, it does jump a little bit. At 5, it jumps, at 7, it jumps, 8 and 9 are not primes, they're prime powers. They jump, 11 is a prime, 13 is a prime, 16 is a prime power. Now, you notice these jumps are not all equal, but it's still very similar to the idea of pi of x. It's jumping at things that are related to primes. And the amazing thing is that look at what it looks like it's doing except for the sort of stair step up and down, it's varying around this dotted line, but the dotted line that seems to give the idea, it looks like a straight line. That's why psi of x is such a nice function. So, it's very similar to just prime counting in a normal way, but it turns out that if you count them in a sort of smarter way with logarithms, it really looks like it's going up in a straight line, and that's a nice thing. It's a simple kind of graph, simplest kind of graph you could have. So, the idea is in fact that the prime number theorem from this prize of the 19th century basically is to show that this special way of counting primes, the psi way of counting primes, if you put in 1,000 for example, you basically just get 1,000. Or if you put in a million, you get a million, very simple. So, now we need to take it off in a little bit of a different direction. We now have a statement as to where we'd like to go on the way of this, a conjecture about the distribution of primes, but how would you start attacking this? The problem here is that the main techniques in mathematics have to do with algebra and formulas, things like x squared and the quadratic formula and all kinds of stuff like that. And it seems like there's no formulas for these guys. It's just counting and the yes or no questions, is this prime or not? There doesn't seem to be any algebra. So, Euler, going back a little bit into the 18th century, one of the only mathematicians who could rival Gauss, he attacked what seems to be a totally unrelated problem, and I don't want to talk about. It's called the Basel problem. Basel was, you know, the city in Switzerland where a lot of these folks were hanging out. And the problem was, what is the number I get by taking one and then adding a fourth to it and then a ninth and then a sixteenth, et cetera? And the question is, wait a minute, first thing you might think of, isn't that obviously just infinity? How could I possibly add an infinite number of numbers to each other and not get infinity? And I need to check the timing on this here. Okay, I checked the timing. Now, isn't this obviously just infinity? I'm adding an infinite number of things to each other. How could I possibly not get just an infinite number? So, I want to first go back a little bit and compare it to a couple of other very, very famous infinite sums. One is one plus a half plus a third plus a fourth plus a fifth, all the reciprocals of all the integers. Add those all up. Another famous one is add one and a half and a fourth and an eighth and a sixteenth and a thirty-second, all the reciprocals of the powers of two. These are the kind of things you'd see on a ruler, you know, like halves, fourths, eighths inch, sixteenth inch, thirty-second inches. Our system, the old-fashioned English-American system likes those things a lot. Let me do the second one first. There's a beautiful, beautiful fact. It's called the geometric series. It's a beautiful way to see that this actually is going to give us a finite answer amazingly enough. So, say that one plus a half, another way to write that is two minus a half. This is three halves, this is three halves. Now, if I add a quarter to that, what I can do is instead of taking two and subtracting a half, I can just subtract a quarter. So, that this is going to, it turns out this is seven fourths, this is seven fourths. This guy, one plus a half plus a fourth plus an eighth, well, let's see. Instead of subtracting a fourth, I really should just subtract an eighth. If you look at the picture here, this is a famous picture, it's that if what I'm trying to do is I'm trying to get from zero to two on the number line. And what I do is I go halfway. And then I go halfway, half of the remaining distance. And then I go half of that remaining distance. So, when I get like one plus one half, for example, it's two, but then minus the one half I haven't covered. One plus a half plus a fourth is two minus the fourth I haven't covered. Then if I add an eighth, it's two minus the eighth I haven't covered. Or then two minus the sixteenth I haven't covered. And so, if I look at all these points, one, one plus a half, one plus a half plus a fourth, one plus a half plus a fourth plus an eighth, et cetera, all those points, they're not getting anywhere close to infinity. They're never, ever bigger than two. And in fact, they get closer and closer and closer and closer to two. And in fact, what mathematicians say is that we define the infinite sum to be the trend of these numbers. That it's true that any one of these finite sums never quite gets to two. But that's not what we're trying to define. We're trying to define what would it be to add them all up. And we say we're going to define that to be exactly the number two. It turns out to make a heck of a lot of sense, even though it seems a little weird to give ourself permission to do that. So the question is, oh, maybe all of these sums are finite. Well, let's think about it for a sec. If I added one plus two plus three plus four plus five, let's look down here for a second. Like one plus two plus, if the numbers are actually getting bigger, that's clearly going to be infinite. There's no way to make sense of that finitely, or no simple way, anyway. But let's look at the harmonic series. That's the other one I was saying was very famous. One plus a half plus a third plus a fourth plus a fifth. All the reciprocals of the integers. There's a very clever way to analyze that one. And it gives a very different answer. It's actually pretty much to relate it to the geometric series. To realize that the half, fourth, eighth were so special. If I just replace the three with a four here, I'm definitely making it smaller. So the original one that I'm analyzing, by replacing three with a four, this guy is bigger than this guy. The first three terms here are bigger than the first three terms here. Let's leave this one fourth alone. And now I'm going to replace all the five, six, seven with eights. That makes these fractions smaller than these guys. And so this one that I'm trying to analyze is definitely bigger than this guy. Then I'm going to take everything through 16. And I'm going to replace them with sixteenths. And then continue. Then it's going to be 30 seconds, 64th, et cetera. But let's see. We can group these naturally. That's one. That's a half. A fourth plus a fourth is another half. Ooh, I've got exactly four one-eighths. That's a half. I've got eight sixteenths. That's a half. And so now I'm adding one plus a half plus a half plus a half plus. If you look at that on the number line, it's just moving to one-half, moving by one-half to the right over and over and over again. I'm not slowing down and stopping like I was here by adding smaller and smaller. I keep adding the same amount. And that's definitely infinite. Now that doesn't tell us exactly how fast this goes off to infinity because this was actually bigger than this guy. But if you've got this guy and it's bigger than this guy, and it goes to infinity, then the original one must go to infinity, okay. And so that'll be a good place to stop for this section. But let me just say, in the next section, we're going to come back to the Basel problem. And it turns out that's one of the ones where it is finite. It's sort of in between the harmonic and the geometric. But it turns out that it was known to be finite. But the question was, what is its sum equal to?