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  • TONY: I've got another constant for you Brady.

  • It's Apéry's constant.

  • One point two oh two oh five six nine ...

  • We could go on.

  • What is this really? This is actually - what I'm writing down here -

  • is really 1 + 1 over 2 cubed + 1 over 3 cubed ...

  • This sum actually appears in quantum electrodynamics.

  • It's related to the anomalous magnetic moment of the electron, which is one of the great tests of QED.

  • This number appears in that so it has physical relevance,

  • but that's not really why we're going to talk about it today.

  • The reason I want to talk about it is because it's -

  • It's a crazy number in many ways

  • and not a lot was known about it for a very long time, and it's still not known about.

  • This goes back to 1644 when Pietro Mengoli posed the Basel problem.

  • Of course, the Basel problem was - the question was not quite this sum

  • but essentially this sum

  • 1 plus 1 over 2 squared, 1 over 3 squared and so on...

  • To get this in closed form - this was the problem, to get it in closed form

  • and this was famously solved by Euler in 1735

  • and he showed that this was equal to π squared over 6.

  • Now of course, Euler being Euler,

  • you know, wasn't going to settle for just this guy, right?

  • He wanted to do this far more generally

  • so he looked at the general form of these things: 1 over 2 to the s, 1 over 3 to the s

  • 1 over 4 to the s and so on which we recognize as the

  • Riemann Zeta function, you know, one can ask what values this takes in the positive integers?

  • And this is what Euler was interested in and he was able to show of course that when you take even integers -

  • so you look at this guy at even values - that this took the form of something rational.

  • times π to the 2n

  • So of course, π being transcendental, this guarantees that this number is also going to be transcendental.

  • That was for the even positions.

  • What about the odds? The things like zeta to the 2n + 1 and which of course

  • This guy which is of course

  • our Apéry's constant is at the value three. What about this? Is this rational? Is this transcendental?

  • What, what is it? Euler couldn't work it out, Euler couldn't do it and nobody knew for hundreds of years

  • What was going on with this number.

  • Until this mathematician came along, a kind of maverick mathematician -

  • a good mathematician, but not a great mathematician, a guy called Roger Apéry.

  • actually solved this problem.

  • And Apéry's quite an interesting character actually:

  • He's French, he was very political, very politically active

  • He was a prisoner of war in the 2nd World War

  • and he sort of had quite a dry sense of humor actually.

  • Later on he was a member of the French resistance.

  • At one point he was actually stopped by the Gestapo because he was carrying an object -

  • A long thin object wrapped in newspaper.

  • And the Gestapo said to him: what are you carrying there? Is it a gun?

  • And he said no, it's a leg.

  • And it was a leg, it was his friend's prosthetic leg, that he was going to get fixed

  • So he was there, but you know, this was the Gestapo he was messing with, right?

  • So he was pushing his luck somewhat.

  • Anyway this sort of sense of humor - this sort of maverick style - actually carried through.

  • Let's fast-forward to 1978 - he's coming towards the end of his career

  • there's this - he's a good mathematician, but not a great one,

  • and he gives a seminar about the value of this zeta function at three, okay?

  • And whether it's rational or not which hadn't been and which the great Euler himself had not been able to solve.

  • Anyway, Apéry says he solved it. Okay. It is irrational, and he can prove it

  • So he starts his - and everyone's a bit skeptical, okay?

  • So he starts his seminar and the first thing he writes down is the following.

  • Doesn't really matter, details of that don't matter

  • The point is nobody in the audience has ever seen this formula before, okay?

  • It's not being proven, nothing.

  • So they immediately ask, you know, where did - no one's seen this before - where did they find it?

  • Where did he find this formula? And he says - he replies - "Oh, they grow in my garden."

  • So everybody's getting annoyed at this point. People start to leave and like, you know, everyone's furious,

  • and then while this is all going on - people are leaving -

  • some guy at the back's got a programmable calculator,

  • and manages to verify this sum, at least to the accuracy of the calculator,

  • and then, you know, as the talk's going on, he interrupts and says and announces this,

  • at which point everyone sits up and starts paying attention.

  • And it turns out that Apéry has solved this hundreds of years old problem,

  • and he proves that it is indeed irrational.

  • And of course it is fabulous - it sort of shook maths at the time.

  • People started to try to use it to then start to ask questions about the other odd values - you know -

  • zeta 5, zeta 7, and so on.

  • And they try to use the same method, and all methods like that have so far failed,

  • interestingly enough, and so it's still not known whether those guys are irrational or not.

  • BRADY: But three we cracked. TONY: Three is definitely irrational,

  • but it's not known whether it's transcendental.

  • What is known about the other odd ones is that there are an infinite number of irrational ones,

  • but we don't know which ones they are.

  • It's even known that one of zeta 5, zeta 7, zeta 9, and zeta 11,

  • that one of these is irrational, but we don't know which one.

  • BRADY: (laughs) TONY: So it's so little known about it and

  • It's sort of - I think it's a fabulous number.

  • So coming back to Apéry - I did a shout-out on Twitter.

  • I asked people to send in three randomly chosen positive integers,

  • and the reason I did that was 'cause I wanted to estimate Apéry's constant.

  • Now I'm a bit lazy, so I didn't go through all of them 'cause I've got loads!

  • And some people put ones that were a little bit hard to manipulate as well.

  • There's quite some quite entertaining comments about how you can't choose that random numbers from just 140 characters

  • And so on and so forth, but anyway

  • I did go through someone I went through quite a few.

  • I have a list here, and what I did was I went through those numbers,

  • and I checked whether the combinations were co-prime. So in other words I asked

  • What was their greatest common divisor - of the three numbers -

  • and if that greatest common divisor was 1, then they're co-prime.

  • If it was some number higher than 1 then they're not co-prime.

  • So I just went through and I checked them all, all the different combinations.

  • So let me do this - I've gotta count this now, Brady!

  • I haven't actually checked this - I didn't check the number - [if] the estimate came out right,

  • so this is real-time, real-action stuff.

  • So anyway the number that are co-prime -

  • Let's just - let's just count them, okay

  • So here's my list, so that's one - all the ticks are co-prime, okay.

  • BRADY: These are all the tweeters?!

  • TONY: These are all the tweeters - not all of them 'cause there were just too many, so I had to stop at some point,

  • but it's a lot of them, okay?

  • So these are all the tweeters - so anyway, okay,

  • So if they're ticked, then they're co-prime - they sent in co-prime combinations. So one, two,

  • three, four, five, six, seven, eight, nine,

  • ten, eleven, twelve, thirteen, fourteen, fifteen, ...

  • 38, 39, 40, 41, 42,

  • 43, 44, 45, ... 68, 69, 70, 71, 72.

  • So there's 72 of those came in as co-primes, co-prime combinations.

  • Some people didn't send in co-prime combinations

  • What I've written down here

  • are the greatest common divisors that they had so we'll just count how many of those there were.

  • One, two, three, four, five, ... fourteen, fifteen.

  • All right, did I miss any?

  • Okay, so that's a total of - it was 87 in total, right?

  • Okay, now I claim, that -

  • This is the moment of truth now, I dunno if this is going to work -

  • That 87 divided by the number of - so the total number divided by the co-prime number - let's see what number this gives.

  • (excited laughing)

  • It's not bad, not bad!

  • Is that good?

  • BRADY: It's pretty close. TONY: I think it's pretty close. I'm happy with that.

  • I'm very happy with that estimate. Okay, good.

  • Yay, and I think, I'm sure if I'd gone through all of them, this would've got more and more accurate, right?

  • Okay, how? Why?

  • Let's suppose we had s numbers.

  • What's the probability that any given number - so we got some prime number p -

  • What's the probability that any given - you pick a random number - that it's divisible by p?

  • It's 1 over p. If I've got s numbers, what's the probability that they're all divisible by p?

  • Well that's going to be 1 over p to the s.

  • What's the probability that at least one of those s numbers is not divisible by p?

  • That's 1 minus 1 over p to the s.

  • So if they - if I now ask,

  • whether if I've got s numbers,

  • whether the probability that their lowest [*greatest] common divisor is 1,

  • the basis - there's no number of the prime numbers that divides them all,

  • then I just have to take the product of this number over all the primes.

  • So that's the product over the primes

  • of 1 minus 1 over p to the s and now our friend Euler comes back, right?

  • Because it's all Euler - he's always [like?] that, Euler - Euler showed that the zeta function

  • Was the product over the primes

  • 1 minus 1 over p to the s to the minus 1.

  • So this is 1 over zeta of s.

  • Okay so this means you've got a 1 in zeta s chance of finding a co-prime combination.

  • Okay, so for three -

  • I've got three numbers. The chances of finding a co-prime combination are 1 in zeta 3.

  • That's why it worked. Makes sense?

  • (outro music)

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  • We'll see you in the next video!

TONY: I've got another constant for you Brady.

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アペリの定数(Twitterで計算) - Numberphile (Apéry's constant (calculated with Twitter) - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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