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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
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here's that final animation what the
derivative of the zeta function looks like