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.

A lot of people know about this function

because there's a one-million-dollar

prize 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... 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

enthusiast who started to read 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

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)

+ 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

value like : s = 2

you get 1

+ (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

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

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

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

of negative 2 = 0, but when you plug

in -2 it gives you

1 + 4 + 9 + 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

one-half to the real part times one-half

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 like

1/2 to the power of 2 + i which

is one-half squared times one-half to

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

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 gonna 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

to learn more about complex functions

you have no heart.

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

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

1 + 2 + 3 + 4 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 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

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

entirely about leveraging the properties

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

know about this extension is 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

information about prime numbers it's

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

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

another thing we know about this

extended function is 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

on and on up to infinity equals

-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

instead it comes from analytically

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

admit that the uniqueness 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

supported in part by audible.com which

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here's that final animation what the

derivative of the zeta function looks like