Euler and the Basel problem
The Basel problem asks for the sum of the reciprocals of all perfect squares: 1 + 1/4 + 1/9 + 1/16, continuing to infinity. Euler solved it, showing the answer is π²/6, using a method that was considered hand-wavy until Weierstrass proved it rigorously about a century later. More importantly, Euler generalized the problem into the zeta function, which sums reciprocals of natural numbers raised to any power s, and showed that this sum over natural numbers equals an infinite product over the prime numbers alone. The chapter closes by introducing the harmonic series (s=1, which diverges) and the geometric series as two tools needed to show why that prime-product connection works.
- The Basel problem is named for the city associated with Euler and the Bernoulli family, a dynasty of mathematicians who attempted the problem before Euler and failed.
- The Bernoulli family is also behind Bernoulli numbers and the Bernoulli principle, which is commonly but incompletely cited as the explanation for how airplanes fly.
- Euler introduced the zeta function in a text called 'Various Observations of Infinite Series'.
- The harmonic series, where s=1, does not converge; it grows without bound, which is what makes s=1 a special and important boundary case for the zeta function.
Transcript
This chapter, from the episode video's captions · 1,058 words
10:29The problem is pretty simple. Okay, you take an infinite series of reciprocals of squares and you add it up. So you get 1 over 1 + 1 over 2^2 + 1 over 3 2 + 1 over 4^2 all the way up to infinity. This is a infinite series. So 1 + 1 over4 + 1 over 9 + 1 over6 all the way. What does that equal? This was proposed in the 1600s. It's named the basil problem because the people who were really working on it were Leonard Oiler and the Bernoli family. The Bernoli family is the very famous family that is behind like the Bernoli numbers um the Bernoli principle which is how a lot of people mistakenly say airplanes fly.
11:11It's not that simple. I mean it has something to do with it but it's not that simple. Um, in any case, the Bernoli family tried and they failed. This is a family of mathematicians. Leonard Oiler tried. Honestly, might have been his first time that he just like solved it. Okay? And he said, "This whole thing is equal to p<unk>^2 over 6." Okay, so 1 + 1 over 4 + 1 over 9 + 1 over 16 all the way down. That's going to equal<unk> 2 over 6. The way he did it was kind of hacky. Um, and like it was only proven to be completely correct by wire strauss like a 100 years later. >> It was a little handwavy. >> It was a little hand wavy, but he got the right answer. Like so many things that he did. Okay, cuz his intuition was
11:53just like insane, right? Okay, fine. So that's impressive, right? But that's not all his main thing that Oiler did for the basil problem. So the basil problem is only 1 plus 1 over 4 plus 1 over 9 all the way over, right? um Oiler in his text various observations of infinite series he generalizes to something called the zeta function. Okay, he actually invented the zeta function which is what if I consider not just one over squares but one over cubes >> or one over uh stuff to the fourth power things [clears throat] like that. So he he creates this function called the zeta
12:33function. And crucially he shows that that sum of one over all of the squares and things like that of the natural numbers is equal to a product over the primes. >> Oh. Okay. So so he took something that was discrete as an initial problem set >> generalized it and came up with and had a rule that related to that generalization. >> Very good. Yeah. The generalization was I take one over the cubes or whatever power that I want. And the little trick that he did was show that the sum could be turned into an infinite product >> over primes. We're going to get into exactly how that works because I think this is a really neat way of seeing why
13:16the zeta function is so important in mathematics. It has to do with primes. Okay? And everywhere you'll look, you'll see that the remon hypothesis has to do with the distribution of primes. Understanding just this will give you kind of a sense of why that is the case. Why are primes related to this infinite sum? Yeah. Okay. So that's what we're going to do right now. Um here's where we're going to start. We're going to start with the zeta function. Let's consider instead of um squares. So in this case, the zeta function is for any arbitrary s. >> Yes. >> But the s is usually two. That'll give you the basil problem of 1 + 4 1 over 4 1 + 1 9th + 1/16th. Instead, let's just consider s= 1. Okay, this is called a
13:57harmonic series. It's just 1 plus all of the fractions added up. This thing crucially does not converge. Okay, this thing goes to infinity. >> So there's the harmonic series which is when the power is one. There's the basil problem when the power is two and then >> and then the zeta is for any power greater than one. Okay, that's the zeta series. Mhm. Okay. So, we're going to consider the harmonic series, which is when the power is one. So, we're just getting reciprocals. Okay. We're going to try to understand this a little bit. Now, we're going to need a second tool in our toolbox, and that has to do with the equation for a geometric series. A geometric series is a power series. It's something like suppose you were to add 1
14:37+ 1/2, but not not instead of 1/3, we say plus 1/4 >> plus [clears throat] 1/8. So powers of two, the same power over and over again. Okay, this is not now you're not changing the number. >> Yes, >> you're now change you're you're keeping the number the same, but the power is going up. >> The uh the variable moves from being in the power spot to being in the the >> exponent in the exponent [clears throat] spot into the >> Yeah, exactly. Exactly. So So now this is a geometric series is what it's called because there's a common ratio between each of the terms, right? You're halfing each term and then you're adding it up. So you get a one, that's a whole pizza, let's say. Then a half, that's half a pizza plus a quarter of a pizza
15:18plus an eighth plus a 16th plus a 32th or whatever it's called. >> 32. [laughter] >> As you add it all up, you're going to get two. You can see, right, the one is the first, and then as you add a half plus a fourth plus an eighth and so on and so forth, you're going to get a whole number and the whole thing is going to equal two. the way, and this is something that you um learn in high school calculus, is how to sum up a geometric series. You take the common fraction, which is a half, sorry, the common ratio, which is a half, cuz you're multiplying a half every time you add a new thing. Um, and you take the
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