Hello, everyone!
A lot of people have been asking me
to explain one of the great
unsolved problems in mathematics. It's called the Riemann Hypothesis
Now, the fact that a lot of people have been asking me
to explain this tells me two things.
First, my life is weird.
And two, that you're a bunch of mercenaries,
because you've probably heard
that the Riemann Hypothesis
is one of those million dollar problems--
that if you solve it
you will win one million dollars.
Now it is one of the great unsolved problems in mathematics.
It was number eight on David HIlbert's list
of challenges to 20th century mathematicians.
It was number one on a similar list
of challenges to 21st century mathematicians,
and it is one of these Clay Institute million dollar prize problems.
If you've heard of it at all,
you may have heard that it's something to do with prime numbers,
but you don't know exactly what.
So I'm going to attempt to explain it to you,
although it's not going to be easy!
They don't give away million dollar prizes for school maths.
But if you want it, the 'too long/did not watch' version is
The Prime Number Theorem.
The Prime Number Theorem approximates how many primes there are
less than a given value, x.
Now it is an approximation--
it's not the true value.
There is going to be a difference from the true value,
an error.
Now, the Riemann Hypothesis will tell you
how big that error term is.
If you want a little bit more information,
the error term that we're looking for
is related to a function called
the Riemann zeta function,
and particularly when that function is equal to zero.
So the Riemann Hypothesis is
the zeroes of the Riemann zeta function
all lie on the line
1/2 plus i t,
So these are complex numbers,
but with real part equal to a half.
Now, to explain this hypothesis,
I'm going to go into the Prime Number Theorem,
in a little bit more detail,
I'm going to tell you what the Riemann zeta function is,
how it's related to primes,
why the zeroes are important,
and if I have time,
I will prove that black equals white,
and cat equals dog.
So let's start with the Prime Number Theorem.
The Prime Number Theorem was first thought of by Gauss
at the end of the 18th century, when he was only 15 years old.
You'd think he'd be busy texting,
and hanging around the outside of shops,
but no, he's busy doing the Prime Number Theorem.
Now, if we call the number of primes less than x,
if we call that pi of x,
then the Prime Number Theorem says that
pi of x, divided by x over log (x),
tends to one, as x tends to infinity.
So some people say that pi of x 'twiddles' [~] x over log (x).
You can think of that as pi of x,
the prime counting function,
is approximately x over log (x).
So if we do an example of that,
if we say x is 100,
then the number of primes less than 100
there are 25 of them.
But our approximation,
x over log (x),
gives you 22,
which is not bad, really, for an approximation.
In fact, the error gets proportionally smaller
the larger the numbers are.
So if I used x as a billion,
then the number of primes
less than a billion
is something like 50 million.
And our approximation,
x over log (x),
gives me 48 million.
It's about 2 1/2 million off
from the true answer.
But proportionally, it's a smaller error.
Forty years later, Dirichlet and Gauss
came up with an even better approximation
for pi of x.
They said the pi of x 'twiddles' [~] this integral:
between 2 and x of 1 over log (t) dt.
It's called the logarithmic integral,
and it is an even better approximation.
So, for example,
if I had x was a billion, again,
then the logarithmic integral
actually gives me an approximation
of about 50 million.
It's, in fact, only 1,700 away
from the true value,
so it's a really good approximation for pi of x.
Now, there is another quantity which I'll mention now,
because it's going to turn up later.
We've seen pi of x, which is the prime counting function.
You get one point for each prime.
I'm going to introduce now
something called J of x,
which is the prime power
counting function.
Again, you get one point for each prime,
but you also get 1/2 a point
for each prime squared.
So something like 9,
which is 3 squared--you get
half a point.
You get a third of a point if it's a prime cubed.
So 27 scores a third of a point, because
it's three cubed.
And you get, for the nth power of a prime,
you get one over n points.
So that's the prime power counting function.
Remember that,
because that is going to turn up later.
You can see it's related to pi,
using this formula,
and you can reverse that--
you can actually get pi
from J, using this other formula,
which has some more
minuses and pluses in it.
But, the prime number theorem
was actually only proved 100 years after
Gauss thought of it.
And it was finally proved
using the Riemann zeta function.
The Riemann zeta function was actually
based on something Euler was doing
in the 18th century.
You see, Euler--
he was interested in infinite sums, like
one, plus a half, plus a cube, plus a quarter--
and that goes on forever.
If you've seen that before, you may know
that's called the Harmonic Series,
and it is tending to infinity,
very slowly--
it's tending off to infinity.
What Euler was interested in were powers of those fractions.
So one plus a half squared,
plus a cube squared,
plus a quarter squared,
and that going off to infinity.
And he found that that doesn't tend
to infinity--
that actually tends to a number.
The number is actually
pi squared over 6.
So if that's the first time you've seen that, hooray!
Surprise pi!
If you do it for cube powers,
one plus a half cubed,
plus a third cubed,
that actually tends to
1.202...
if you do it for fourth powers,
you get pi to the 4 over 90, and so on.
So you actually get proper values,
except for when the power is one.
The Harmonic Series--that tends off to infinity.
In general,
Euler was studying these
sums of one over n to the power s.
Now what he found was
that this infinite sum
was equal to an infinite product--
an infinite product of primes.
If p is your prime,
then it's a product of p to the s,
over p to the s minus one.
So you go through the primes, like
2 to the s over 2 to the s minus 1,
multiply that by
3 to the s over 3 to the s minus 1,
multiply that by the next prime--
5 to the s over 5 to the s minus 1,
and that goes on forever too.
Now what it showed was
if, say, s is equal to 2 there,
then that infinite product of primes
is equal to pi squared over six.
Or if s was equal to one,
then that infinite product of primes
goes off to infinity,
like the Harmonic Series goes off to infinity.
which is actually a proof that there are
infinitely many primes
because if there were only finitely many primes,
you'd get a finite number.
Now, it's the Euler function
that Riemann decided to rip off--
I mean, study.
Bernhard Riemann was apparently
a shy, timid individual,
but mathematically very talented.
Gauss was actually his lecturer at university
It was Gauss who persuaded Riemann
to swap his course from theology
to maths,
and in the end,
Riemann ends up running the place.
He becomes head of department in the end.
In was in 1859, though,
that Riemann wrote his paper on the zeta function.
Now, it's one of the most important papers in number theory,
it's only ten pages long.
Riemann never wrote on number theory before or since that paper.
This was his one and only time
and it just turns out to be
one of the most important papers in number theory, ever!
It's like the mathematical equivalent of a mic drop.
Peace, out.
I'm done.
Riemann's zeta function was basically
Euler's function
except he used it for all
complex numbers.
So s could be any complex number
not just one and two and three and four.
The problem with that is
it is only defined
when the real part of s
is greater than one.
Otherwise, it explodes--
it goes off to infinity,
like the Harmonic Series.
What Riemann needs is a function
that's defined for all complex numbers,
but is equal to the Euler function
when the real part is greater than than one.
Now, the amazing thing about complex analysis is
there is a way to do that, and there is only one way to do it.
So if you think about it, what it means is
if you know a value of a function for some complex numbers,
you automatically know it for all complex numbers.
Which is like [pow]
What Riemann did
is he considered this
infinite sum.
It's like the Euler function,
but you've got alternating plus and minus signs in it.
And he divided it by this factor,
1 minus 2, to the power 1 minus s,
and this function is defined
for complex numbers where the
real part of s is greater than zero.
And, it's equal to the Euler function
when the real part of s is greater than one.
This is the function we're looking for!
Now, this defines
a value when the real part of s
is positive,
what about when the real part of s is negative?
To do that,
what Riemann did next was
he considered that strip,
between zero and one,
and in that strip, we have this
true identity for the zeta function.
It's not very nice to look at,
I know--
so you've got the zeta of s
equals--some junk.
And it's got sin in there,
and its gamma, and
and pi--it's a lot of junk.
multiplied by zeta (1 - s).
What this means is
the values are kind of reflected
in the real part of s equals a half line.
They're reflected, and then multiplied by that horrible factor.
So that means we can reflect
all our positive values
into our negative values.
And that will define it.
So if you want to know zeta of minus one [-1],
then you would multiply it by that horrible factor,
with s equals minus one,
which is only minus one over two pi squared,
then you would times that by zeta of 2.
Now we know the value of zeta for 2,
if you remember that from before-
that was pi squared over six.
So all together,
zeta of minus one is minus one-twelfth. [-1/12]
Now, if we consider the zeta function
as an extension of the Euler function,
then some people consider the sum
one plus two plus three plus four,
going on forever,
some people consider that to be
equal to minus one-twelfth. [-1/12]
And in a way, it kind of is.
So to recap, the Riemann zeta function comes in three parts:
where the real part of s is greater than 1,
then it's just equal to the the Euler function,
which you can write as a product of primes, as well.
When the real part of s is between zero and one,
then it's equal to this alternating sum,
and when the real part of s is negative,
well, you take the positive value,
reflect it in the real part of s equals a half line,
multiplied by a horrible factor.
And that defines it all,
Anything I haven't defined,
you can define by taking limits.
So zeta of zero is equal to minus a half [-1/2],
zeta of 1 tends off to infinity,
just like the Harmonic Series did.
The next question is, when is this function equal to zero?
Well, for a start,
we know that when the real part of s
is greater than one, it's never equal to zero.
because it's a product of primes.
And none of those factors are zero.
So what about the third case, when it's negative?
Well, in the third case,
you might be able to see some zeroes at minus two,
minus four,
minus six,
and so on,
because we've got a sin function in there,
and that's when the sin function is zero.
Those zeroes are called 'trivial' zeroes,
Otherwise, we've got
non-trivial zeroes,
and they exist in the strip
between zero and one.
And they're also symmetric around the real part of s
equals a half line.
So Riemann wondered if his zeta function
could be written in terms of the
non-trivial zeroes.
This is like what you do with a polynomial.
So imagine you have a polynomial like
x cubed plus 2x squared minus 13x plus 10
well, you know you can factorize that
into three factors,
because it's a cube.
And you find the factors
by looking for when the polynomial
is equal to zero.
For example, if x equals 1,
then the polynomial equals zero.
So x minus 1 is a factor.
If you do that again, if x equals 2,
the polynomial is equal to zero,
so x minus 2 is a factor.
And if x equals minus 5,
the polynomial is zero,
so you can say x plus five is a factor.
And that's how you factorize a polynomial.
So Riemann wondered if he could
do something like that
with the zeta function.
and you can!
So he worked out that the zeta was equal to
again, some junk,
you've got gamma still in there,
and pi is still in there--multiplied by a product of factors
involving the non-trivial zeroes.
that's one minus s over rho.
Rho are your non-trivial zeros.
So you can do that.
So on one hand, you've got zeta,
which is your normal zeta function,
which was defined with its three parts,
On the other hand, you've got zeta written as a product
of its non-trivial zeroes, and those two things are equal.
So you do--you equate them,
and then you just mess around with it.
You take logs of both sides,
you do some calculus,
some complex analysis,
some theorems or residues,
it's all very nasty stuff,
I'm not going to do it now,
but in the end, you end up with a result involving
J of x.
That's the prime power counting function, if you remember.
And that is equal to
the logarithmic integral of x,
you may remember that from the Prime Number Theorem--
plus a sum of logarithmic integrals of x to the power rho.
Rho are your non-trivial zeroes, so they're still there.
minus log of two.
That's just a constant--that's still there because
there was a factor of a half, and that's still there.
Your pi's have disappeared--they've cancelled out and disappeared--
and your gamma has turned into this integral
which is not very nice looking,
and also not very important--don't worry about it.
So we have a kind of a prime number theorem now
for J of x.
the prime power counting function,
We can now say J of x
'twiddles' [~] the logarithmic integral.
And we have an expression for the error, the difference--
if we subtract J of x minus the logarithmic integral,
is equal to this error term.
The most important term in the error term
is that sum of logarithmic integrals
of x to the rho.
If rho are your non-trivial zeroes,
they are really controlling the size
of that error term.
Now your non-trivial zeroes are symmetric
around the real part of s equals a half line.
If all your non-trivial zeroes
actually lie on the real part of s equals a half line,
then that error term
will be as small as possible.
Nearly forty years after Riemann and
100 years after Gauss first thought of the Prime Number Theorem,
mathematicians were finally able to prove it.
And the proof involved the Riemann zeta function.
In particular, they showed that the zeta function
has no zeroes on the line
real part of s equals one. And that did it!
That proved the Prime Number Theorem!
Hooray!
Now, the error term
for the Prime Number Theorem
well--that's controlled by the non-trivial zeroes
of the zeta function--the Riemann Hypothesis.
And that still isn't proven, so
boo!
But if the Riemann Hypothesis is true,
then the error term has a bound.
It has to be less than c square root of x, log of x.
for some constant c.
So it can't be as big as it wants to be.
The Prime Number Theorem also tells us that the average gap between consecutive primes
should be about log(x).
And if the Riemann Hypothesis is true,
then there is a similar bound on consecutive primes.
Other consequences of the Riemann Hypothesis are things like
the weak Goldbach conjecture.
The Weak Goldbach Conjecture says that
any odd integer greater than five
can be written as the sum of three primes, and that would be true
if the Riemann Hypothesis is true.
Another consequence--one I like--
is if the Riemann Hypothesis is true,
then there will always be a prime between consecutive cubes.
Now the reason I like that
is that means you can define a constant theta
such that theta to the power three to the n,
will always generate a prime number for all the values of n.
It's called a Mill's Constant.
They already exist, actually, but you have to use x as a prime.
Now, if Riemann Hypothesis is true,
then x doesn't just have to be a prime.
And then you would get lots more Mills' Constants,
the smallest of which would be 1.30637 something something.
So I like that one!
There's a lot of evidence for the Riemann Hypothesis
we've found lots of zeroes--trillions of zeroes
on that critical line--that real part of s equals a half line,
Unfortunately, Godfrey Hardy showed that
there are infinitely many zeroes
on that line.
This is evidence--but none of it is conclusive evidence, I'm afraid.
Now, if you want to win
that million dollars,
you have to prove the Riemann Hypothesis.
Unfortunately, if you find a counterexample,
if you find an example of a zero
which is not on the critical line
the real part of s equals a half line,
and you disprove the Riemann Hypothesis--
no million dollars!
No million dollars for you, no!
There is no million dollars if you disprove it by counterexample.
So I'm afraid, mathematicians,
it is now over to you!
And if you have been,
thanks for watching.