Analysis of a Complex KindWeek 6

Lecture 5: The Prime Number Theorem

Petra Bonfert-Taylor

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 1 / 9

The Power of Complex Analysis

Complex analysis is an extremely powerful field. This is demonstrated forexample, by the ability to prove a deep theorem in number theory, the PrimeNumber Theorem, using complex analysis.

Well describe this in more detail in this final lecture.

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 2 / 9

The Prime Counting Function

Let pi(x) = number of primes less than or equal to x . This function is called theprime counting function. Example:

pi(1) = 0pi(2) = 1pi(3) = 2pi(4) = 2pi(5) = 3pi(6) = 3pi(7) = pi(8) = pi(9) = pi(10) = 4pi(11) = pi(12) = 5

It seems impossible to find an explicit formula for pi(x). One thus studies theasymptotic behavior of pi(x) as x becomes very large.

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 3 / 9

The Prime Number Theorem

pi(x) = number of primes less than or equal to x .

Theorem (Prime Number Theorem)

pi(x) xln x

as x .

Note: The symbol means that the quotient of the two quantities approaches 1as x , i.e.

pi(x)x/ ln x

1 as x .

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 4 / 9

A Brief History Of pi(x) x/ ln xEuler (around 1740) discovered the connection between the zeta function(s) (for real values of s) and the distribution of prime numbers.60 years later, Legendre and Gauss conjectured the prime number theorem,after numerical calculations.Another 60 years later, Tchebychev showed that there are constants A,B(with 0 < A < B) such that A

xln x pi(x) B x

ln x.

In 1859 Riemann published his seminal paper On the Number of PrimesLess Than a Given Magnitude. In this paper he constructed the analyticcontinuation of the zeta function and introduced revolutionary ideas,connection its zeros to the distribution of prime numbers.Hadamard and de la Valle Poussin used these ideas and independentlyproved the Prime Number Theorem in 1896.The main step in their proofs is to establish that (s) has no zeros on{Re s = 1}.

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 5 / 9

How is (s) Related To Prime Numbers?

Euler discovered:(s) =

p

11 ps ,

where the (infinite!) product is over all primes.Here is why this is true:

(s) =11s

+12s

+13s

+14s

+15s

+16s

+17s

+

=

(1 +

12s

+14s

+18s

+ )(

1 +13s

+19s

+ )(

1 +15s

+1

25s+

)

=p

k=0

1pks

=p

11 1ps

.

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 6 / 9

The Riemann Zeta Function and Prime Numbers

(s) =p

11 1ps

Note:This product formula shows that (s) 6= 0 for Re s > 1.The key step in the proof of the prime number theorem is that has no zeroson {Re s = 1}.The details of the proof of the prime number theorem go beyond the scope ofthis course.The prime number theorem says that pi(x) x

ln x, but it doesnt have any

information about the difference pi(x) xln x

!

However, the prime number theorem can also be written as pi(x) Li(x),where Li(x) =

x2

1ln t

dt is the (offset) logarithmic integral function.

Lecture 5: The Prime Number Theorem Analysis of a Complex Kind P. Bonfert-Taylor 7 / 9

The Riemann Hypothesis and Prime Numbers

The proofs of the prime number theorem by Hadamard and de la VallePoussin actually show that pi(x) = Li(x) + error term, where the error termgrows to infinity at a controlled rate.Von Koch (in 1901) was able to give best possible bounds on the error term,assuming the Riemann hypothesis is true. Schoenfeld (in 1976) made thisprecise and proved that the Riemann hypothesis is equivalent to

|pi(x) li(x)|