Name: リーマン仮説の鍵 - Numberphile (The Key to the Riemann Hypothesis - Numberphile)
Uploaded: 2021-01-14T10:15:49.000Z
Duration: 12 min 38 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

yes so we're going to talk about L-functions. I want to give you an example

様態の副詞

first of all of one L-function that I think you probably would recognize and

this is the Riemann zeta function, which depends on a complex number

s is x+iy. 'i' here is the square root of -1, and you can define it either by

a sum over the integers, all the integers, all the positive ones. Or, this is a great discovery of Euler:

I can write it as a product over the prime numbers. and so the zeta function

encodes information about the primes. It contains all the primes and if you can

unravel how the information in the zeta function is encoded you can determine

properties of the primes. That's one of the L-functions, and let me tell you

a little bit more about the Riemann zeta function because that's gonna be part of the

story. So the Riemann zeta function is defined as a sum over the integers or a

product over the primes, and these formuli work if x is greater than one.

But it turns out you can find formuli that match these when x is greater than 1.

When x is less than one work as well. And here's what the Riemann zeta function

looks like. So these formuli work to the right of the line x=1. So they work

out here. And they tell you the value of the zeta function anywhere in this in this

region. Now it turns out the zeta function has this amazing symmetry

which was discovered by Riemann, which say that

the line that I have drawn in blue there that passes though the point one half, that's

a symmetry line of the Riemann zeta function. If I know the value the zeta

function at some point here, I reflect that point through the blue line to get

a point here and know its value here. I can deduce that very easily. So Riemann reflection

formula, the symmetry formula, tells us that because I know he zeta

function in this range I also know in this range as the reflection so that

just leaves this strip which we don't understand. This trip is where the

function is really mysterious. We know that in this strip,

there are infinitely many points where the zeta function takes the value 0.

The belief is - and this is the Riemann hypothesis - that these points where the

takes the value zero, all lie exactly on the symmetry line, and this is called the

Riemann hypothesis. And this is one of the great mysteries of mathematics.

Riemann hypothesized this in 1859 and it's been tested hugely since.

BRADY: professor, it's well-known in mathematics that if any mathematician can prove the

Riemann hypothesis, he or she is destined for greatness.

PROF:  Well, because the Riemann zeta function encodes information about the

prime, so if we know the positions of the zeros, that tells us really important

BRADY: What information? What does it tell you that you don't already know?

PROF: it tells us things we already suspect, because we sort of

believe the Riemann hypothesis is true, but there are things we can't prove. So for example,

how many primes are there up to a trillion? We have a formula that we believe is true but we

can only prove that formula if we know the Riemann hypothesis is true.

BRADY: So it's like a whole bunch of houses would suddenly have a foundation. PROF: Absolutely,

and there are whole books written assuming the truth of the Riemann hypothesis,

so the foundations of our understanding of the primes would disappear if the Riemann

hypothesis turned out to be false. It's known there are infinitely many zeros of the

Riemann zeta function inside the critical strip, and it's known that at

least 40% of those do lie on the symmetry line. It is known, and this is a

result of hugely extensive computations that the first 10 trillion zeros

lie exactly on the symmetry line. That's the first ten thousand billion. It's known that

batches of zeros, much higher up, beyond the 10 trillion, lie on the line,

so I think the world record at the moment is there's a batch of zeros up beyond the 10^36th zero,

That is the billion billion billion billion-th zero, somewhere up there

there's a whole batch which we know lie exactly on the line, so people have put a

pretty huge effort into this, and there's a long history of this:

Alan Turing, when he built the first electronic computer, one of the first things he did

was compute the zeros of the zeta function. He found about the first 1000 lie on the line.

BRADY: professor, if a mathematician one day finds a zero in the strip but not on the line,

is that mathematician gonna be a hero or a pariah?

PROF: [laughs] Well they would certainly be very famous, and I think mathematicians

very sanguine about this, they'll just accept the truth, they want to know the

truth, that person would be famous, but I think it would be a kind of

ugly kind of fame, not a beautiful kind of fame.

So we got these properties and this is what I want to emphasize: we can write the zeta

function as a sum over the integers, or as a product over the primes,

and the zeta function has a symmetry line, reflection symmetry line,

and we believe that all the zeros of the zeta function around that line

lie exactly on that. This is one of the great mysteries in mathematics, and people have

spent 150-odd years thinking about this. And one natural way

if you're given a mystery is to try to understand: well, is this a more general

property? So are there other functions which look like the Riemann zeta function?

And maybe by finding lots of those functions and comparing them that tells you the

essential properties that make the Riemann hypothesis true. Maybe it allows

us to prove it. So an example would be: if you think about evolution,

Darwin wanted to establish evidence for evolution, so he goes to the Galapagos Islands,

and he has to find lots of finches that're all very similar but not

identical, and by making comparisons of the similarities and the differences you

then understand the essential properties that led them to evolve as they did.

So we want to find the cousins of the Riemann zeta function. To see whether

there are any, what might you do? Well you might look at this sum over the integers,

and you might change some of these plus signs into minus signs and see, well,

can you have any combination of pluses and minuses here? And it turns out

it's very rare that by changing some of these plusses into minusses you can write

this sum, the resulting sum, as a product of the primes

but be slightly different to this product but would resemble it. and you'd have something

that had a symmetry line that's very very rare that that happens. to there is

one example of one. let me go back to the Riemann zeta function going to split the

primes up into two classes: 2 will always be special believe that is it is

will either be - if you divide them by 4 - either remainer of 1 or 3.

7: divide that by 4, uh the remainder is 3.

where the remainder's 3 will flip their sign, so we put that one there

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リーマン仮説の鍵 - Numberphile (The Key to the Riemann Hypothesis - Numberphile)

essential

pattern

associate

evolve