Riemann Zeta Function
First Meeting: The Prime Number Theorem
Fall 2004
The prime numbers are the positive integers (the set of positive integers is also called the set of natural
numbers and is usually denoted N) that have exactly two different factors. So the list of prime numbers
starts out 2, 3, 5, 7, 11, . . . . Since every natural number larger than one can be written in a unique way as
a product of primes the prime numbers can be thought of as the basic building blocks of all other natural
numbers. This way of decomposing any natural number into a product of primes is considered so important
that it usually referred to as the Fundamental Theorem of Arithmetic.
Theorem 1 (Fundamental Theorem of Arithmetic) Every natural number n > 1 can be written in
the form n = p1 e1 p2 e2 · · · pk ek where the pi are distinct primes and the ei are natural numbers. This representation is unique except for the order of the primes.
There are many famous questions in mathematics that involve the prime numbers. Here are a couple.
1. How many primes are there? There is a proof in Euclid’s book The Elements written around 300 B.C.
that there are an infinite number of primes.
Theorem 2 There are an infinite number of primes
Proof: Assuming the number of primes is finite and the list p1 , p2 , . . . , pn contains all the
primes, create a number that contradicts the FT of A.
2. What is the frequency of the prime numbers? One way to measure this frequency is to investigate the
than or equal to n
for natural numbers n
ratio number of primes less
n
Theorem 3 limn→∞
number of primes less than or equal to n
n
=0
Proof: something we could look at later
The second question has interested mathematicians for a while it was while investigating this question
that Bernhard Riemann developed the Zeta function. Riemann used an idea of Euler’s from around 1740
and extended it to a new setting. Here is some back ground information surrounding the context in which
it was developed.
Define π(x) = the number of primes less than or equal to x. The values and growth rate of π(x) record
something about the distribution and frequency of the prime numbers. In 1798 Joseph Legendre wrote a
book “Essai sur la Theorie des Nombres” in which he presented evidence that π(x) is approximately equal to
x
was published posthumously in 1863,
ln(x)−1.08366 . At roughly the same time Karl Gauss determined, which
Rx u
x
that π(x) could be approximated by ln(x) and more accurately by 2 ln(u) du, called the logarithmic integral
and denoted LI(x). However these were unproven conjectures based since neither Legendre or Gauss could
back up there statements with proofs. In 1850 P.T. Tchebychef proved that there exist positive constants a
and b, with a < 1 < b, such that a(x/ ln(x)) < π(x) < b(x/ ln(x)) whenever x was large enough. However,
he could not determine values for a and b and not determine how large x had to be. However he was able
to prove that IF the limit
π(x)
lim
x→∞ x/ ln(x)
existed then this limit must equal 1. Proving that this limit existed became known as the Prime Number
Theorem. In 1859 Riemann addressed this question in a paper called ” On the Number of Prime Numbers
less than a given Quantity”. In this paper he introduced the Zeta function, made some conjectures about
this function and about how it related to the prime number theorem. Finally, in 1896 Hadamard and de la
Valle Poussin used Riemann’s ideas to independent of each other to prove the prime number theorem.
In order to understand Euler’s Zeta function we need to review some ideas from calculus.
Infinite Series:
P∞
PN
• Convergence
of n=0 an . Let SN = n=0 an and consider limN →∞ SN . If the limit exists then we
P∞
say n=0 an converges. If not, or if the limit is ±∞, we say it diverges.
• Formally: Given > 0 there exist number L and M such that whenever N > M then |SN − L| < • Some well known series
– Geometric Series if |x| < 1 then the series
–
ex =
P∞
n=0
sin x =
n=0
xn =
1
1−x ,
or 0
∞
X
x2n+1 (−1)n
(2n + 1)!
n=0
–
cos x =
• Convergence of
P∞
∞
X
xn
x2
x3
=1+x+
+
+ ···
n!
2!
3!
n=0
–
–
xn is convergent. In fact,
∞
X
x2n (−1)n
(2n)!
n=0
∞
X
xn
ln(x + 1) =
(−1)n+1
n
n=1
P∞
n=0 cn (x
− a)n (Pages 423,424 of Calc Text. NOTE: choice of M depends upon x)
• Uniform convergence of a series. Choice of M is independent of x.
• Defining a function by an infinite series. Define the Euler Zeta function ζ(s),
ζ(s) =
∞
X
1
ns
n=1
for every natural number s > 1
P∞
Problem 1: Show that for all s > 1, the series n=1 n1s is convergent. This will show that ζ(s) is a
real number for every whole number s > 1
P∞
Problem 2: Show that n=1 n11 is divergent. This will show that ζ(1) is undefined.
• Definition of an infinite product and convergence of infinite products
Q∞
QN
– Convergence
Q∞ of n=0 an . Let PN = n=0 an and consider limN →∞ PN . If the limit exists then
we say n=0 an converges. If not, or if the limit is ±∞, we say it diverges.
– Formally: Given > 0 there exist number L and M such that whenever N > M then |PN −L| < • Startling Connection (Euler 1740): For all s > 1,
ζ(s) =
Y
p a prime
Euler’s argument for this
ps
s
p −1
=
Y
p a prime
1
1 − p1s
!