The Prime Number Theorem
Anusha Pai
Outline (Table of Contents)
Prime Numbers - brief summary of prime numbers and its properties.
Historical Introduction of The Prime Number Theorem - summary of where the theorem originated and its
evolution
About the Prime Number Theorem - brief description of what the Prime Number Theorem is
History of Selberg and Erdös - Selberg and Erdös life before the Prime Number Theorem ’era’
Dispute Between Selberg and Erdös - discussion on the conflict of who gets the credit to the Prime Number
Theorem
Outline of Newman’s Short Proof on the Prime Number Theorem - brief summary of Newman’s life and his
short proof
Outline of Selberg’s Proof
Part That Is Going To Be Explained
Notations Explained - notations that are necessary to understand before understanding the last part of the
Elementary Proof of the Prime Number Theorem
Preliminaries - concepts that are necessary to master before dealing with the explanation of the last part of
the proof
Proof - proof of the last part of the proof
Remark - overall conclusion of the paper
Prime Numbers
Prime numbers are a fascinating mathematical topic. In fact, it is the most mysterious among most mathematical concepts. Prime numbers are numbers that are not divisible by other than one and itself. Now what
does this mean? Well, let’s do some examples.
2 is a prime number because it can be divided by 1 and 2 and not have a remainder.
6 is not a prime number because although it is divisible by 1 and 6, it is also divisible by 2 and 3. Therefore,
six is not a prime number.
Prime numbers are mysterious because till this date nobody has discovered a pattern for prime numbers.
There are patterns for even and odd numbers, but none for prime numbers. The largest prime number found
is 25 7, 885, 161 and that too from a computer. The gap between prime numbers in the number sequence is
sometimes enormously large and other times it is the number right next to another prime number.
Because of Euclid, we now know that there are infinitely many prime numbers. Because of his findings, the
Prime Number Theorem was able to be true. Euclid is best knwn for his textbook on geometry called The
Elements. He is called the ”Father of Geometry” because of his great findings. In his textbook, he has a list
of axioms or set of rules. Some of the axioms mentioned were:
-a straight line is formed when given two points -when given a center and a radius, a circle can be formed -all
right angles are equal
His discoveries were called Euclidean Geometry. Also in his textbook, he discussed that there are infinitely
many primes numbers. This prompted a huge change in mathematic history on prime numbers.
Euclid explained that if we say that there are finitely many primes, then it would contradict the fact that
a number after a number is not divisble by the previous number. The product of the ’finitely’ many prime
numbers is a prime number and the number after the prime number is not divisible by the previous prime
number or the prime numbers in the product. This contradicts the notion of divisibility.
Historical Introduction of The Prime Number Theorem
After Euclid’s proof showing that there are infinitely many primes, the fame of prime numbers increased.
Legendre and Gauss were the first ones to come up with the conjectures of the Prime Number Theorem
independently. In fact, the Prime Number Theorem improves that Euclid is correct that there are infinitely
many primes.
Legendre’s expression:
lim
x→∞
π(x)
x
log x
=1
Gauss’ expression:
π(x)
lim R x dt = 1
x→∞
2 log t
Gauss’ expression is called the Li function (Li(x)).
Gauss and Legendre were two brilliant mathematicians. Gauss, at the age of ten, was asked to find the sum
of all the whole numbers from one to one hundred by his teacher. The rest of his classmates were adding each
and every number while Guass found a pattern. He noticed that the ends equal 101. Meaning, 1 + 100 = 101
and 2 + 99 = 101 and so on. He then multiplied 101 by 50 and got 5050 as his answer. For his answer, he
did not have any work, just the number. His teacher was impressed by his mathematic ability. Although his
family was against him furthering his studies, his relatives noticed his talent and got the opportunity to excel
and take forward his talent. The expression of the Prime Number Theorem was one of his contributions to
the mathematical world.
Anyways, back to the two expression. We know that the two expressions are equivalent because if we take the
limit of the Legendre’s expression over Guass’ epression and use L’Hopital’s Rule, we eventually get 1. This
means that the two limits are growing at the same rate and therefore are equivalent.
This was the beginning the Prime Number Theorem. However, Chebyshev cracked the problem open by
coming up with another equivalent expression using ψ.
Chebyshev set
ψ(x) =
X
pk ≤x
and got the expression
2
log p
ψ(x)
=1
x→∞ x
lim
which is equivalent to
θ(x)
=1
x→∞ x
lim
where
θ(x) =
X
log p
p≤x
These expressions was the start of the elementary proof of the Prime Number Theorem. In fact, Chebyshev
should be given credit for the PNT because he gave the first clues on how to prove the theorem.
The first ones to prove the Prime Number Theorem were Jacques Hadamard and Charles de la Vallée-Poussin
and Selberg was the first to prove the elementary proof of the theorem.
Donald Newman found the shortest proof of the Prime Number Theorem. This is discussed more in depht in
the section Shortest Proof of the Prime Number Theorem.
About The Prime Number Theorem
Now what exactly is The Prime Number Theorem? What are we trying to understand?
The main idea is that we want to show that π(x) and
x
grows at the same rate.
log x
π(x) is the number of primes before the number x. In mathematical notation, this is:
π(x) = #{p ≤ x} =
X
1
p≤x
Now to show this, we know that if we take the limit as x goes to infinity of the ration of the two functions π(x)
x
and
and it equals one, then that proves the Prime Number Theorem. It may seem like an easy concept,
log x
but it really isn’t. Selberg shows an elementary proof of the Prime Number Theorem
History of Selberg and Erdös
Paul Erdös and Atle Selberg are the masterminds of the 20th century. Erdös was born on March 26, 1913 in
Budapest, Austria-Hungary. He was born to Anna and Lajos Erdös, both mathematic teachers, and was the
only child who survived after age 5. His interest in mathematics started at a young age, specifically at age 4.
He would ask random people for their age and would calculate the number of seconds they have lived mentally.
(wikipedia)
Atle Selberg was born on June 14, 1917 in Langesund, Norway. He was born to Anna Kristina Selber and Ole
Michael Ludvigsen Selberg. His mother was a teacher and his father was a mathematician. Selberg lived during
World War II which, in a way, forced him to work alone on his mathematics. Germans invaded Norway so he
could not collaborate with anyone. This impacted his working style, which is discussed later on. (wikipedia)
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At age five, Erdös ”realized that 250 less than 100 is 150 below zero,” and at age seven, Selberg ”showed the
difference between consecutive squares is an odd number.” The two mathematicians were in fact like North and
South. They were total opposites.
Erdös loved working with other mathematicians and he would always work around a large amount of mathematicians. He had about five hundred coauthors. On the other hand, Selberg preferred working alone, taking
his own time. Selberg likes working alone because of World War II. During this time, he had to work alone and
this method of working stuck with him and he preferred it later too. In fact, Selberg has only collaborated with
one other mathematician named Chowla, and it was because Chowla had a question for Selberg. Not only are
their work preference opposite, but according to Tim Gower’s essay ”The Two Cultures of Mathematics,” their
mindsets on their mathematics are different. Gower explains that there are two point of views on mathematics:
1. ”The point of solving problems is to understand mathematics better.”
2. ”The point of understanding mathematics is to become better able to solve problems.”
According to Gower, Erdös is group 1 and Selberg is in group 2. If we think about it, it does make sense.
Erdös prefers working with a good amount of people and when one works and discusses with a lot of people,
one develops a better understanding on the topic. On the other side, since Selberg likes working alone, he
has a better chance of developing as a mathematician. By using his mind alone, he was able to be a better
problem solver and this allows him to understand mathematics better.
In terms of work, Erdös worked with combinatorial and probabilistic number theory, combinatorial geometry,
probabilistic and transfinite combinatorics and graph theory. Selberg has dealt with postive density of zeroes
of the Zeta-function which has lead Weyl be fond of Selberg. This is very important during the Prime Number
Theory Controversey. Selberg was best known for his analytic number theory (wikipedia).
Dispute Between Selberg and Erdös
In March 1948, Selberg proved his fundamental formula:
θ(x) log x +
X x
θ
log p = 2x log x + O(x)
p
p≤x
Erdös noticed that this formula is the access point to all proofs.
In April 1948, Selberg proved that a + A = 2, and he knew that the Prime Number Theorem (PNT) could be
proved easily after that.
Selberg proved that a + A = 2 in a simple manner.
He assumed that
lim
x→∞
π(x) log x
=1
x
a is the lower limit and A is upper limit. If the upper limit and the lower limit equals one, then the whole
limit equals one.
Therefore, since a and A equals one, the limit equals one and
a+A=2
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From May to July 1948, Selberg discussed Dirichlet’s theorem and briefly mentioned his fundamental formula
to Turan. Then Turan gave a seminar on Selberg’s Dirichlet’s Theorem proof and mentioned the fundamental
formula to Erdös.
pn+1
In August 1948, Erdös could noticed that he could derive
→ 1 from Selberg’s inequality. pn refers to the
pn
nth prime. For example, the first prime is 1, the second prime is 3, the third prim is 5, and so on.
Selberg disliked how Erdös was working on his fundamental formula.
pn+1
→ 1, Selberg found his first proof
pn
of the PNT, but it was rather long. The proof was proven using Erdös’ result that in the interval (x, x(1+δ))
x
there are more than c(δ)
>primes for x>xo (δ).
logx
On September 1948, while Erdös was proving as n approaches infinity,
Selberg tries to discourage Erdös to prove his result. Selberg says that ”Erdös’ result was obtained entirely
independent of [his] work.”
The dispute really began when it came to a joint versus separate paper for the elementary proof of the PNT.
In August 1948, Selberg writes to Erdös that he will publish his proof of the PNT as it is and will mention
that he used Erdös result. Selberg tells Erdös to publish his result separately in which it does not mention the
PNT.
Later, Selberg says tha the will publish a second proof and asks Erdös to publish his findings, but Erdös
disagreed with it because Erdös felt that he would have proved the PNT a long time ago if Selberg told his
about a and A.
According to Joel Spencer and Ronald Graham’s article, ”Weyl rejected Erdös’ paper and published only a
version of Selberg’s paper and avoided Erdös’ contibutions.” Later Erdös published his paper and Erdös won
the Cole Prize while Selberg won the Fields Medal.
Outline of Newman’s Short Proof on the Prime Number Theorem
The short proof, but more complex proof of the Prime Number Theorem was founded by Donald Newman.
Donald Neuman was born in Brooklyn, New York on July 27, 1930 and died on March 28, 2007. He was a
brilliant mathematician. He participated in the Putnam Math Competition. The Putnam Math Competition
is basically where many students come on the first Saturday of December and take an examination consisting
of undergraduate mathematic material. However, the material being tested on is not basic Calculus. It is
more complex. The problems on the exam are more advanced and one needs ot be a very good problem
solver. Newman was the Putnam Fellow for three straight years which is incredible. He was friends with
John Nash, another amazing mathematician who has had the opportunity to have his life in a film called
Beautiful Mind. Newman recieved full scholarship at Harvard for a Ph.D program. According to the Journal
of Approximation Theory, his main focuses on mathematics were classical analysis, approximation theory,
number theory, combinatories, functional analysis, and recreastional mathematics. Newman worked for the
government and was a professor for many prestigious unversities. His short proof of the Prime Number Theorem
was published in his textbook called Complex Analysis.
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Newman proves properties of the three functions listed below where ζ is the Riemann Zeta function and p is
a prime number.
1)
ζ(s) =
∞
X
1
ns
n=1
2)
Φ(s) =
X logp
p
ps
3)
θ(x) =
X
log p
p≤x
where s C and x R
Newman then proves an Analytic Theorem where he proves that:
∞
Z
f (t)dt
0
exists.
This is a brief outline of his proof. Keep in mind that it includes many complex ideas. It looks so elegant yet
simple, but it is difficult to understand for an average student with basic Calculus knowledge.
Outline of Selberg’s Proof
Below is a general outline of the steps that are needed to prove the Prime Number Theorem.
Before we begin, we know:
θ(x) =
X
log p
p≤x
and
π(x) = #{p ≤ x} =
X
p≤x
It is enought to show that the
lim
x→∞
which means
θ(x)
−1=0
x
θ(x) − x
=0
x→∞
x
lim
Selberg defines that:
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1
R(x) = θ(x) − x
P N T ⇐⇒ lim
x→∞
R(x)
=0
x
The Prime Number Theorem (PNT) is:
lim
π(x)
x→∞
x
logx
However, Selberg proves
lim
x→∞
R(x)
=0
x
which is equivalent to the above notation of the Prime Number Theorem.
Selberg creates an averaging formula and if this formula is proved, then the Prime Number Theorem is therefore
proven. This type of method is called a Tauberian Argument (proving an averaging formula to show that a
theorem is indeed true).
The averaging formula is:
X x
log p = 2x log x + O(x)
θ(x) log x +
θ
p
p≤x
Selberg shows with his relation and partial summations:
1 X x
log log x
|R(x)| ≤
( ) +O x
log x
n
log x
n≤x
We know that x
log log x
is infinitely slower than x.
log x
lim
x→∞
log log x
logy
= 0 ⇐⇒ y = logx, lim
=0
y→∞ y
x
Now, Selberg shows that there exists a number K > 0 [a big number K], so that for every δ > 0 [a very small
K
K
number δ] and every x > e δ , the interval (x, xe δ ), so that for every z in it, we have that
|R(z)|<4δz ⇐⇒
Now the bootstraping method begins.
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R(z)
<4δ
z
θ(x)<(4 log 2)(x)
With this, we will show that R(x) < ax where a = 4 log 2 + 1 which also mean that a<8.
By considering the inequality,
x
log log x
1 X
R( ) + O x
|R(x)| ≤
log x
n
log x
n≤x
R(x)
is especially small, we can obtain
x
a2n
x
that by going far enoiugh the initial inequality |R(x)|<ax can be improved to |R(x)|<a 1 −
300K
a2n
If we define a1 = a(= 1 + 4 log 2) and an + 1 = an 1 −
, then an approaches zero, which then solves
300K
the Prime Number Theorem.
along with the fact that we have those special intervals over which
Part That Is Going To Be Explained
Since the iteration-process
an+1 = an (1 −
a2n
)
300K
K5
obviously converges to zero if we start for instance at a1 = 4 (one sees easily that then an < √
), this proves
n
(4.1) and thus our theorem.
Notations Explained
an : this means the nth term of a series. For example, if an = 2n, then then a5 is (2)(5), which is 10.
an+1 : this is similar to the above notation; however, rather than the nth term, it is the term after that. So
a5+1 = (2)(6), which equals 12.
K: K is a variable that is greater than 1. It is actually a very large number. A number that can be as large
as you desire. Therefore, the fraction will be less than one (explained in the proof).
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Preliminaries
Convergence: If a function converges, then its limit as x goes to positive infinity reaches a value a. Specifically,
for a series to converge, the series has to be positive and decreasing, then only the series will converge to a
number.
For example:
∞
X
n2
n=1
does not converge because although the series is positive, it is not decreasing.
∞
X
1
n2
n=1
converges because all the terms are positive and the terms are decreasing.
Sequence: A sequence is a set of values that have a distinct pattern.
Proof
The following are the assumptions:
1.
|R(x)|<ax
for a < 8 for some interval x>xo
2.
0<δ<1
where x>eK/δ and (x, xeK /δ) contains a subinterval in the form of (y, yeK/2 ) where
R(z)
<4δ
z
Now we have to show that for every
a2
x1 >0 : |R(x)|<a 1 −
300K
x
where x>x1
This means that our estimates get improved successively according to the latter a1 = 4 an+1
= an 1 −
a2n
.
300K
It is enough to show that an approaches 0.
However, to achieve this, we first need to show that an is positive and decreasing. We can show that it is
decreasing through induction. If we know it is decreasing and positive, then we can using limits to prove that
an converges.
a2n
We know that
is always less than one because K is a massively large number and any number over a
300K
very large number will equal less than one.
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For example,
1
10
1
100
1
1000
1
10000
1
100000
1
1000000
= 0.1
= 0.01
= 0.001
= 0.0001
= 0.00001
= 0.000001
As we can see, if the denominator is large, then the result is always less than one.
If
a2n
a2n
is less than one, then 1has to be less than one too.
300K
300K
We know that a1 is less than 8 and we know that K is greater than 1 [K is a huge number]. With this
information, we can imply that an is decreasing because if we go further and further, an is getting smaller
and smaller. We Know that an is positive because an is bounded below by 0. Therefore, an is decreasing and
positive.
Now we can show that an converges.
By taking the limit of both sides,
l =l 1−
l2
300K
the two l’s cancel and we have:
l3
=0
300K
The denominator is very large and when we take the limit where the denomitor is very large, the limit tends
to zero.
In conclusion, the Prime Number Theorem is proven.
Remarks
Overall, Selberg’s elementary proof of the Prime Number Theorem is amazing and beautiful. The incorporation of Calculus, primarily logarithms, developed the proof nicely. Although there was a dispute between
Selberg and Erdös, both Selberg and Erdös have made their mark in mathematic history. When it comes to
mathematics, it is all about who achieves the credit, but in reality, all mathematicians are brilliant. Every
discovery and every proof is important. Selberg and Erdös come from different backgrounds, but they are both
important mathematicians.
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References
1. http://kobotis.net/math/MathematicalWorlds/Fall2015/131/Projects/PNT/ErdosSelberg.pdf
2. http://www.math.technion.ac.il/hat/people/obits/newman.pdf
3. http://kobotis.net/math/MathematicalWorlds/Fall2015/131/Projects/PNT/ErdosSelbergDispute.pdf
4. http://kobotis.net/math/MathematicalWorlds/Fall2015/131/Projects/PNT/Rourke.pdf
5. https://www.math.hmc.edu/funfacts/ffiles/20001.5.shtml
6. https://en.wikipedia.org/wiki/Atle Selberg
7. https://en.wikipedia.org/wiki/Paul Erd%C5%91s
8. http://www.britannica.com/biography/Euclid-Greek-mathematician
9. http://www.math.wichita.edu/history/men/gauss.html
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