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#### 可视化黎曼zeta函数和解析延拓

Visualizing the Riemann hypothesis and analytic continuation

The Riemann zeta function.

This is one of those objects in modern math that

a lot of you might have heard of, but

which can be really difficult to understand.

Don’t worry, I’ll explain that animation

that you just saw in a few minutes.

out for anyone who can figure out

when it equals 0. An open problem known as

the Riemann hypothesis. Some of you may

have heard of it in the context of the

divergent sum 1 + 2 + 3 + 4…
+…听说
on and on up to

infinity.

You see there’s a sense in which the sum

equals -1/12, which seems

nonsensical if not obviously wrong. But a

common way to define what this

equation is actually saying uses the Riemann zeta function.

But as any casual Math

into this knows its definition references this one

idea called analytic continuation which

has to do with complex-valued functions and this idea can be frustratingly opaque and unintuitive,

so what I’d like

to do here is just show you all what
ζ函数究竟长什么样
this zeta function actually looks

like and to explain what this idea of

analytic continuation is in a visual and

more intuitive way. I’m assuming that you

know about complex numbers and that you’re comfortable working

with them, and I’m tempted to say that

you should know calculus since analytic continuation is

all about derivatives but for the way I’m

planning to present things I think you might actually be fine without that.

So to jump right into it let’s just define

what this zeta function is for a given input where we

commonly use the variable’s’ the function

is 1 over one to the’s’ (which is always 1)
（也就是1） 加上1/(2^s)
+ 1 over 2 to the’s’ + 1 over 3 to

the’s’ + 1 over 4 to the’s’

on and on and on

summing up over all natural numbers.

So for example let’s say you plug in a
s=2代入其中
value like: s = 2

you get 1
/4+1/9+1/16+…
+ (1 over 4) + (1 over 9) + 1/16 and as

you keep adding more and more reciprocals

of squares this just so

happens to approach pi squared over 6

which is around 1.645 there’s a very
π在这里出现 背后有个漂亮的原因
beautiful reason for why pi shows up here

and I might do a video on a later date

but that’s just the tip of the iceberg

for why this function is beautiful.

You can do the same thing for other inputs’s’

like three or four and

sometimes you get other interesting values and so far everything feels

pretty reasonable you’re adding up smaller and smaller amounts and these sums approach some number… Great,

no

craziness here! Yet if you were to read

about it you might see some people say

that zeta of negative 1 equals
-1/12 但是看看这个无穷级数
-1/12 But looking at this infinite sum

that doesn’t make any sense… when you

raise each term to the negative 1

flipping each fraction you get 1

+ 2 + 3 + 4
4+… 也就是所有自然数的和
on an on over all natural numbers and

obviously that doesn’t approach anything certainly not -1/12, right? And,

as any mercenary looking into the Riemann

hypothesis knows this function is said

to have trivial zeros at negative even numbers

so for example that would mean that zeta
(-2)等于0
of negative 2 = 0, but when you plug

in -2 it gives you

1 + 4 + 9 + 16
+16+…
on and on, which again

obviously doesn’t approach anything much

less 0, right? Well we’ll get to negative

values in a few minutes but

for right now let’s just say the only thing that

seems reasonable this function only makes sense when’s’ is

greater than one which is when this sum

converges so far it’s simply not defined

for other values.

Now with that said Bernhard Riemann was

somewhat of a father to complex

analysis which is the study of functions that

have complex numbers as inputs and outputs.

So rather than just thinking about how

this sum takes a number’s’

on the real number line to another number on the

real number line his main focus was on understanding what

happens when you plug in a complex value for’s’,

so for example maybe instead of

plugging in 2, you would plug in 2 + i

now if you’ve never seen the idea of

raising a number to the power of a complex value you

can feel kind of strange at first because it no longer

has anything to do with repeated multiplication but mathematicians found

that there is a very nice

and very natural way to extend the definition of

exponents beyond their familiar territory of real numbers and into the

realm of complex values. It’s not super

crucial to understand complex exponents for where I’m going

with this video but I think it’ll still be

nice if we just summarize the gist of it here

the basic idea is that when you write

something like one half to the power

of a complex number you split it up as
1/2的实部次方乘以1
one-half to the real part times one-half
/2的虚部次方
to the pure imaginary part we’re good on

one half to the real part there’s no issues there but what about raising

something to a pure imaginary number?

Well the result is going to be some

complex number on the unit circle in the

complex plane as you let that pure

imaginary input walk up and down the

imaginary line the resulting output

walks around that unit circle

For a base like one half the output

walks around the unit circle somewhat slowly but

for a base that’s farther

away from one like 1/9 then as you let

this input walk

up and down the imaginary axis the corresponding output

is going to walk around the unit circle more quickly.

If you’ve never seen this

and you’re wondering what

on earth this happens I’ve left a few links to good resources in the description for here

i’m just going to move forward with the what without the why.

The main takeaway is that when you raise something
2)^(2+i)时
like 1/2 to the power of 2 + i which

is one-half squared times one-half to
2)^i (1/2)^i那
the i that one-half to the i part is

going to be on the unit circle meaning

it has an absolute value of one. So when

you multiply it it doesn’t change the size

of the number it just takes that

one fourth and rotates at somewhere.

So if you were to plug in 2 + i to
ζ函数中
the zeta function one way to think about

what it does is to start off with all of

the terms raised to the power of 2 which

you can think

of is piecing together lines whose length of the reciprocals of

squares of numbers which like I said

before converges to pi² over six

then when you change that input from two

up 2 + i each of these lines gets

rotated by some amount but importantly

the lengths of those lines won’t change

so the sum still converges it just does

so in a spiral to some specific point on

the complex plane. Here let me show what

it looks like when I vary the input is represented

with this yellow dot on the

complex plane where this spiral sum is

always going to be showing the

converging value for zeta of s

what this means is that zeta(s) defined

as this infinite sum is a perfectly

reasonable complex function as long as

the real part of the input is greater

than one meaning the input’s’ sits somewhere

on this right half of the complex plane again this is

because it’s the real part

of s that determines the size of each

number while the imaginary part just

dictate some rotation.

So now what I want to

do is visualize this function it

takes in inputs on the right half

of the complex plane and spits out outputs

somewhere else in the complex plane a

super nice way to understand complex functions is to visualize them

as transformations meaning you look at

every possible input to the function and

just let it move over to the corresponding output…

for example let’s

take a moment and try to visualize something a little bit easier than the

zeta function: say f(s) = s²

When you plug in s = 2 you get 4

so we’ll end up moving that

point at two over to the point at four

when you plug in -1 you get 1 so

the point over here

at negative 1 is going to end up moving over to the point at 1.

When you plug in

i by definition its square is -1

so it’s going to move over here

to negative 1 now I’m going to add on a

more colorful grid and this is just because things are about

to start moving and it’s kind of nice to have

something to distinguish grid lines during that movement.

From here I’ll tell the

computer to move every single point on

this grid over to its corresponding

output under the function f(s) = s²

Here’s what it looks like

I can be a lot to take in so I

‘ll go ahead and play it again and this time

focus on one of the marked points and

notice how it moves over to the point

corresponding to its square. It can be a

little complicated to see all

of the points moving all at once but the reward

is that this gives us a very rich picture for what

the complex function is actually doing and it all happens in

just two dimensions… so back to the zeta

function we have this infinite sum which

is a function of some complex number s and we feel good and happy

about plugging in values of s whose real part

is greater than one and getting

some meaningful output via the converging

spiral cell so to visualize this

function i’m going to take the portion of the grid

sitting on the right side of the complex plane here where

the real part of numbers is greater than one and

I’m gon na tell the computer to move

each point of this grid to the appropriate output

it actually helps if I add a few

more grid lines around the number one

since that region gets stretched out by quite a bit

alright so first of all let’s just

appreciate how beautiful that is

I mean damn that doesn’t make you want

But also this transformed grid is just begging to be

extended a little bit for example let’s

highlight these lines here which represent all of the complex numbers

with imaginary part i or -i after the

transformation these lines make such lovely arcs before they just abruptly

stopped don’t you want to just you

know continue those arcs in fact you can imagine how

some altered version of the function

with the definition that extends into

this left half of the plane might be

able to complete this picture

with something that’s quite pretty well this is exactly what mathematicians

working with complex functions do! They

continue the function beyond the

original domain where was defined now as

soon as we branch over into inputs where
1的复数
the real part is less than 1 this

infinite sum that we originally used to

define the function doesn’t make sense

anymore you’ll get nonsense like adding
+2+3
1 + 2 + 3 + 4
+4+…的荒谬结果
on a non up to infinity

But just looking at this transformed

version of the right half

of the plane where the some does make sense it’s just

begging us to extend the set

of points that were considering as inputs even if

that means defining the extended function

in some way that doesn’t necessarily use that sum of course that

leaves us with the question how would you define that function

on the rest of the plane?

You might think that you could extend it

any number of ways maybe you define an

extension that makes it so the point at say…

s = -1 moves over to
-1/12 但你也可以对它
-1/12 but maybe you squiggle on

some extension that makes it land on any other value I

mean as soon as you open yourself up to the idea

of defining the function differently for values outside that

domain of convergence that is not based

on this infinite sum the world is your

oyster and you can have any number of extensions right?

Well not exactly I mean

yes you can give any child a marker and

have them extend these lines any

which way but if you add on the restriction

that this new extended function has to

have a derivative everywhere it locks us

into one and only one possible extension

I know I know…

I said that you wouldn’t need to know about derivatives

for this video and even if you do know calculus maybe

you have yet to learn how to interpret derivatives for complex

functions but luckily for us there is a

very nice geometric intuition that you

can keep in mind for when I say a phrase

like has a derivative everywhere here to

show you what I mean let’s look back at

that f(s) = s² example

again we think of this function as a

transformation moving every point s of

the complex plane over to the point s²

for those of you who know

calculus you know that you can take the derivative

of this function at any given input but there

‘s an interesting property of that transformation that

turns out to be related and

almost equivalent to that fact if you look at

any two lines in the input space that

intersect at some angle and consider

what they turn into after the transformation they will still intersect

each other at that same angle.

The lines might get curved and that’s okay but the

important part is that the angle

at which they intersect remains unchanged

and this is true for any pair of lines

that you choose!

So when I say a function has a

derivative everywhere I want you to think about

this angle preserving property that anytime two lines

intersect the angle between them remains

unchanged after the transformation at a

glance this is easiest to appreciate by

noticing how all of the curves

that the gridlines turn into still intersect each

other at right angles.

Complex functions that have a derivative everywhere are called analytic

so you can think of this

term analytic as meaning angle
“保角的” 我得承认这并不完全准确
preserving admittedly i’m lying to a

little here but only a little bit a

slight caveat for those

of you who want the full details is that inputs where

the derivative of a function is 0 instead

of angle being preserved they

get multiplied by some integer, but those points are

by far the minority and for almost all

inputs to an analytic function angles

are preserved so when I say analytic you think angle

preserving I think that’s a fine intuition to have

now if you think about it for a moment

and this is the point that i really want

you to appreciate this is a very

restrictive property the angle between

any pair of intersecting lines has to

remain unchanged and yet pretty much any

function out there that has a

name turns out to be analytic the field of

complex analysis which Riemann helped to

establish in its modern form is almost

of analytic functions to understand

results in patterns and other fields of math and science.

The zeta function defined by this infinite sum on the

right half of the plane is an analytic

function notice how all of these curves

that the gridlines turn

into still intersect each other at right angles

so the surprising fact about complex

functions is that if you want to extend

an analytic function beyond the domain

where was originally defined for example

extending this zeta function into the

left half of the plane then if you

require that the new extended function

still be analytic that is that it still

preserves angles everywhere it forces you into only one possible

extension if one exists at all

it’s kind of like an infinite continuous

jigsaw puzzle for this requirement of

preserving angles walks you into one and

only one choice for how to extend it

this process of extending an analytic

function in the only way possible that

still analytic is called as you may have

guessed”analytic continuation” so that’s

how the full Riemann’s zeta function is defined

for values of s on the right

half of the plane where the real part

is greater than one just plug them into

this sum and see where it converges and

that convergence might look like some kind of spiral since raising each

of these terms to a complex power has the

effect of rotating each one then for the

rest of the plane we know

that there exists one and only one way to extend

this definition so

that the function will still be analytic that is so that

it still preserves angles at every single point

so we just say that by

definition the zeta function on the left

half of the plane is whatever that

extension happens to be and that’s a

valid definition because there’s only one possible

analytic continuation notice that’s a very implicit definition

it just says use the solution of this

jigsaw puzzle which through more abstract derivation we know must exist

but it doesn’t specify exactly how to

solve it mathematicians have a pretty good grasp

on what this extension looks like but

some important parts

of that remain a mystery a million-dollar mystery in fact let’s

actually take a moment and talk

about the Riemann hypothesis the million-dollar problem the places where this function equals

zero turn out to be quite important that

is which points get mapped onto the origin

after the transformation one thing we

that the negative even numbers get map to 0 these

are commonly called the trivial zeros

the name here stems from a long-standing

tradition of mathematicians to call things trivial when they understand

quite well even when it’s a fact that is

not at all obvious from the outset we

also know that the rest

of the points that get map to 0 sit somewhere in this

vertical strip called the critical strip

and the specific placement of those

non-trivial zeros encodes a surprising

actually pretty interesting why this function carry so much

information about primes and I definitely think i’ll

make a video about that later on but right

now things are long enough so

I’ll leave it unexplained Riemann hypothesized that

all of these non-trivial zeros sit right

in the middle of the strip on the line

of numbers s who’s real part is

one-half this is called the critical
“临界线” 如果这是真的
line if that’s true it gives us a

remarkably tight grasp on the pattern of

prime numbers as well as many other

patterns in math stem from this now so

far when I shown what the zeta function looks

like I’ve only shown what it does

to the portion of the grid on the screen

and that kind of under sells its complexity

so if I were to highlight

this critical line and apply the

transformation it might not seem to

cross the origin at all however use with

the transformed version of more and more

of that line looks like

notice how its passing through the

number zero many many times if you can

prove that all

of the non-trivial zeros sit somewhere on this line the clay math

Institute gives you 1 million dollars and you’d

also be proving hundreds if not thousands of modern

math results that have already been shown contingent

on this hypothesis being true

that map’s the point -1 over to negative -1/12

and if you plug this into the original

sum it looks

like we’re saying 1 + 2 + 3 + 4
3+4+…等于-
on and on up to infinity equals
1/12 我们要是仍然
-1/12 now they might seem

disingenuous to still call this is a sum since the

definition of the zeta function on the left half of

the plane is not defined directly from this sum

continuing this own beyond the domain

where it converges that is solving the

jigsaw puzzle that began on the right half

of the plane that said you have to

of this analytic continuation the fact that the

jigsaw puzzle has only one solution is

very suggestive of some intrinsic connection between these extended values

and the original sum the last animation

and this is actually pretty cool i’m going to

show you guys what the derivative of the zeta function

looks like but before that it matters to me to

let you guys know who’s making these videos possible

first and foremost there’s the viewers like you supporting directly on patreon

and this particular video was also
.com的赞助 audible
supported in part by audible.com which
.com提供有声书与其他有声读物 我拥有Audible已经
provides audio books and other audio materials actually use audible almost

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life-changer one particularly good book

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due to a very emphatic recommendation of my brother

and it’s one that I think you guys

would like a lot Josh Waitzkin the
·维茨金在他的童年时期一直
author was a national chess champion

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became a world champion in just a few years so

the man knows what it takes to learn and I

found a lot of what he says about

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meaningfully to learning math as well

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here’s that final animation what the

derivative of the zeta function looks like