0:23

This is demonstrated in many different ways, but today I want to show you how it

is demonstrated, for example, by the ability to prove a deep theorem in number

theory, namely the Prime Number Theorem, using complex analysis.

The Prime Number Theorem has, in fact, first been proved using complex analysis.

Nowadays, there are other proofs that are known that don't use complex analysis.

But the first and

most important proof off the Prime Number Theorem is from complex analysis.

We'll describe this in more detail in this final lecture.

To tell me what the prime number theorem says,

let pi of x be the number of primes less than or equal to x.

1:04

This is called the prime counting function.

It counts how many prime numbers there are less than or equal to a number x.

So for example, pi of one Is equal to zero

because there are no prime numbers less than or equal to 1.

Pi of 2 is 1 because there's a prime number, 2.

And that's the only prime number less than or equal to 2.

Pi of 3 is 2 because 2 and 3 are prime numbers less than or equal to 3.

Pi of 4 is still 2 is the same 2 prime numbers that we're counting.

Pi of 5 is 3 because we're gaining an additional prime number now we have 2,

3 and 5 that we're counting.

1:42

Pi of 6 is also 3 because 2, 3 and 5 are the only prime numbers less than or

equal to 6.

Whereas pi of 7 through pi of 10 are 4,

because now I'm counting the prime numbers 2, 3, 5 and 7.

And so forth, so pi of 11 and pi of 12 is 5, and so forth.

So that's a function that seems pretty random if you look at its values.

0, 1, 2, 2, 3, 3, 4, 5.

It seems impossible to find an explicit formula for pi of x.

One therefore studies the asymptotic behavior

of pi of x as x becomes very large.

So for example, one wants to know how many prime numbers are there roughly

that are less than a million.

So the asymptotic formula for pi of x is what one is looking for.

2:31

The prime number theorem says that pi of x is

asymptotically x divided by the logarithm of x.

That symbol, this asymptotic symbol right here means that the quotient of these

two sides namely pi of x divided by x over ln x goes to one as x goes to infinity.

You could also write this as pi of x times

the reciprocal of that quotient so times ln x over x.

So that goes to one as x goes to infinity.

3:13

Euler was the first one to discover the connection between the zeta function,

zeta of s, for real values of s.

And the distribution of prime numbers.

Euler did not look at the zeta function the way.

And later on Riemann did for complex values of s.

He looked at it for real values of s and

discovered its relation to the distribution of prime numbers.

And we'll look at that in a little bit.

60 years after Euler, Legendre and Gauss conjectured the prime number theorem

after numerical calculations had led them to believe it should be true.

Another 60 years later, Tchebychev showed that there are constants A and

B, such that pi of X is bounded above by B times x over ln x,

and bounded below by A times x over ln x.

This does not give you the prime number theorem yet, but

it's closer, it's a step closer to the prime number theorem.

In 1859 Bernard Riemann published his seminal paper,

On the Number of Primes Less Than a Given Magnitude.

In that paper he constructed the analytic continuation of the zeta function that

we spoke about last class and introduced revolutionary ideas connecting

the zeros of the zeta function to the distribution of prime numbers.

4:30

Hadamard and de la Vallee Poussin used these ideas of Reimann's

independently and proved the Prime Number Theorem in 1896.

The main step in their proofs is to establish that the zeta

function has no zeros on the line where the real part s equal to 1.

Remember, the zeta function has.

4:56

These 0s at minus 2, minus 4, minus 6.

Those are so called trivial zeros.

And it is unknown what happens in this strip.

The Riemann hypothesis says that they all were real part of s is equal to one half.

But it is unknown whether there are extra zeroes in this strip.

And both Hadamard and de la Vallee Poussin were able to show that there

are no zeroes on the line where the real part of s is equal to 1.

5:38

So how is the zeta function related to prime numbers?

Euler discovered that zeta of s can be written

as the infinite product over all prime numbers

of 1 divided by 1 minus p to the minus s.

Let me remind you how the zeta function was defined.

6:02

Zeta of s was the sum.

And from 1 to infinity of 1 over n to the s and

we first look to that for s > 1 then we saw we could also do for re part of s > 1.

And Riemann was able extend this zeta function to an analytic

function the entire complex plane with the exception of 1.

6:28

So why is this summation formula the same as this product formula.

Here is why this is true.

Here again is the zeta function.

Is 1 over 1 to the s +1 over 2 to the s, + 1 over 3 to the s, and so forth.

6:45

You could factor this zeta function by the prime factorization s.

1 + 1/2 to the s + 1/4 to the s.

So in this first set of parentheses this

you'll see all the powers of 1 over 2.

So 1 over 2 to the s actually.

7:13

1 over 2, one fourth, one eighth, one sixteenth.

In the next pair of parenthesis you'll see all of the powers of one third.

And the next one is the powers of one fifth and so forth.

7:32

This is an infinite product.

And if I multiply through, I get the 1 over 1 to the s

term by just picking the 1 from each set of parenthesis.

I get the 1 over 2 to s term by taking the 1 over 2 to the s from

the first set of parenthesis and the 1 from all the others.

8:19

I get that by taking the 1/2 to the s and the 1/3 to the s and

other with all the 1 terms again.

So each term is accounted for and I can't get any additional terms because

all I'm doing is When I multiply through here from these products,

I'm going to get a certain number to the power s and

all these numbers occur by the uniqueness of the prime factorization.

I don't get any terms twice.

8:48

Therefore zeta of s is equal to this infinite product,

and in each parenthesis I see a prime number to

the power s, raised to all possible powers k.

So the sum from k equals 0 to infinity of 1 over p to the ks.

9:14

What is the sum k from 0 to infinity, 1 over p to the ks.

Let me rewrite that a little bit.

I can rewrite that as the sum k from

0 to infinity of P to the minus s to the k.

P is fixed, s is fixed.

P to the minus s, p is a prime number, s is a number at least 1.

So p to the minus s is something that is less than 1.

9:49

Therefore this forms a geometric series and I know this series converges.

The value of this series is 1 over 1 minus

the term whose powers we're adding, namely p to the -s.

Using that, I find that zeta of s

is equal to the infinite product over all primes of 1 over 1- 1 over p to the s.

Which is the same as 1 over 1 minus p to the minus s.

That's the way we wrote it up top.

So here is, again, the formula for the zeta function.

Once you have this product formula, it is easy to see that zeta s does not

have any zeroes for a real part of s greater than one.

10:34

And so far we have the summation formula in the mirror.

It was unclear whether something could add up to zero when here's a product.

A product only means zero if one of the factors is zero and

none of these factors are zero and that shows us

that the zeta function is never equal to 0 for real point of s of endpoint.

11:17

The prime number theorem says that pi(x) is asymptotically the same as x / ln x,

meaning again,

that the quotient of these two quantities goes to 1 as x goes to infinity.

But it doesn't say anything about the difference of the two sides,

so pi(x)- x / ln x.

The prime number theorem doesn't say anything about it.

However, the prime number theorem can also be written as

pi(x) is asymptotically equal to Li(x),

where Li(x) is the offset logarithmic integral function,

the integral from 2 to x of 1 over the natural log of t dt.

12:02

Now the proofs of the prime number theorem given by Hadamard and

de la Vallee Poussin actually show that not only is pi(x) asymptotically

equal Li(x), but moreover that pi(x) = Li(x) + an error term.

And they were able to give explicit balance on this error terms,

particularly on the growth rate of the error term.

So the error term does go to infinity as x goes to infinity, but

at a controlled rate, and the control was given by Hadamard and

de la Vallee Poussin in their proof.

Von Koch, in 1901, was able to give the best possible bounds on this error term.

Assuming the Riemman hypothesis is true, Schoenfeld made this precise and

proved that the Riemann hypothesis is equivalent to pi of x minus li of

x being bounded a buff by root of x times natural logarithm of x over eight pi.

Now li of x isn't quite the same as li of x up here, here is a lowercase l.

And that's not a typo.

Up here there's a uppercase L.

So lowercase li of x is the un-offset logarithmic integral function.

So it's the integral of zero to x of 1 over ln t dt.

And it's related to the offset logarithmic integral function,

the Uppercase Li of x is lowercase li of x minus lowercase li of 2.

So this result right here is only true if the Riemann hypothesis is true,

and moreover, it is equivalent to the Riemann hypothesis.

And it tells us a lot about the distribution of prime numbers.

The voracity of the Riemann hypothesis,

therefore implies results about the distribution of prime numbers,

in particular about how regularly they're distributed about their expected

locations and how much they cannot vary from their expected locations.

14:01

So we have reached the end of this course, let me recap what we have learned.

In this course we have learned about complex numbers, the algebra, geometry and

topology in the complex plane.

We learned about complex functions, we studied complex dynamics, we learned about

Julia sets of quadratic polynomials, we studied the Mandelbrot set, we even got

into the conjecture of local connectedness of the boundary of the Mandelbrot set.

We next studied complex differentiation.

We learned about the Cauchy-Riemann equations, about analytic functions.

We studied conformal mappings,

inverse functions, Mobius transformations, even the Riemann mapping theorem.

Then we learned about complex integration and about Cauchy theory.

So we learned Cauchy's integral theorem and

integral formula and their consequences such as Liouville's theorem,

the maximum modulus principle and so forth.

And we finally got to complex theories, we studied power series,

also called Taylor series and we even go to the Riemann zeta function and

its relation to prime numbers, and the Riemann Hypothesis.

I hope you enjoyed this course.