Prime Number Theorem Section 3.3
Emilio Ferral
Prime Number Theorem Intro
Prime numbers are one of mathematics’ biggest mysteries, and they have been for almost as long as numbers
have been identified. In the simplest of terms, a prime number is any number which can only be divided by
itself and the number one and yield a whole number solution, excluding the number one itself. The first few
prime numbers are easy to calculate and list, with the first ten prime numbers being 2, 3, 5, 7, 11, 13, 17, 19,
23, and 29.
As mathematics grew as a field, prime numbers began to gain a tidal wave of curiosity. The main thing people
were curious about was the pattern of growth of prime numbers. To many it was obvious that as one looked
to larger and larger numbers, there were less and less primes, but there was no obvious pattern visible. The
Prime Number Theorem is a theorem which describes how the number of prime numbers begins to decrease
as one takes into account larger numbers, giving a general pattern for the decrease.
Historical Context
Prime numbers have been one of the biggest mysteries in all of human nature. Although the Greeks are the
ones who were credited with being the first to study prime numbers around 2,500 years ago, there is some
evidence that the ancient Egyptians studied primes as well as a bone known as the Ishango bone which suggests
that primes have been studied for over two thousand years.
At this point in time, the majority of our understanding of prime numbers comes from the revelations of the
early Greeks. The largest source of Greek expertise on prime numbers comes from an ancient text known as
E lements, whose construction is credited to Euclid. Little is known about the life of Euclid, but it is known
that he lived during Ptolemy I’s reign. Although Euclid is credited with making the text, not all of the work
is his. Many of the publications included were made by earlier mathematicians, but Euclid had the proper
knowledge to make the collection, and present it in such a manner that would make it easier for more people to
understand. Since the Greeks had such a good understanding of prime numbers, they went on to ask a pretty
obvious question about the pattern of primes This resulted in what is known today as Euclid’s Theorem, which
states that there are infinitely many prime numbers.
Propositions like these, which are all found in E lements, is what modern prime number theory is based on.
Even though E lements laid the groundwork for prime number theory, it left little detail as to how one would
know if a given number is prime or not or what the location of a given prime integer might be. It took
five-hundred years for a mathematician to come up with a simple way to find prime integers. His name was
Eratosthenes, and his method was this: 1) Write out all numbers from 1 to n. 2) Starting with the number 2,
cross out any number divisible by 2. 3) take the next number that isn’t crossed out and repeat the previous
√
step. 4) keep going until out all numbers less than n has been crossed out. The remaining set of numbers
will be all the prime numbers between 1 and n. This method is known as the Sieve of Eratosthenes. This
method is great for finding the first few prime numbers, but for obvious reasons, the larger n becomes, the
more time consuming and difficult this process becomes.
For thousands of years mathematicians have to find a formula which will be able to generate prime numbers.
The main difficulty with this is the randomness of the pattern of growth in prime numbers. Unfortunately, as
the Dark Ages came along, so did the decline in Greek study of the world, and along with that went the study
of prime numbers. Around the 17th century, the first signs of primes being studied once again appeared as
a French mathematician known as Pierre de Fermat began to study the mystifying numbers. Fermat became
interested in one of the propositions from E lements, the proposition states that if (2n − 1) is prime, then
2n−1 (2n − 1) is a perfect number, with a perfect number being a positive integer that is equal to the sum of
its proper divisors. Due to the difficulty of dealing with larger numbers, the greeks were only able to find
the first four perfect numbers, which are 6, 26, 496, and 8,128. With this information Fermat created his
own propositions to help find the next perfect number, although unlike Euclid, Fermat did not provide any
proofs of his work. A theorem which is known at Fermat’s Little Theorem states that if p is a prime and a
is any positive integer, then p divides ap − a and it would take another 100 years before Euler would prove
the theorem. This theorem serves to determine if an input value is prime or not, and the theorem is widely
used in mathematics and computer science. Fermat is also known for trying to expand on Euclid’s theorem
on the relationship between primes and perfect numbers, and as a result he produced a theorem which tests
for primality, which is the one stated previously. Much of the work that Fermat did served to widen our
perspective and knowledge of prime numbers.
A revolution happened when a fifteen year old known at Carl Friedrich Gauss posed a new idea to the
mathematicians around him. His thought was that since primes are random, then it is likely impossible to
predict the location of the next larger prime. So he thought of a different approach, his approach being
attempting to predict the number of primes within a given interval. At the age of fifteen, Gauss was studying
logarithmic tables, and he noticed a very peculiar pattern. He noticed that as the powers of ten increased, the
ratio of primes increased by approximately 1.7. Here is a table to illustrate the pattern.
x
π(x)1
x/π(x)
10
4
2.5
102
25
4
103
168
5.5932
104
1229
8.137
105
9592
10.425
1010
455,052,511
21.975
1015
29,844,570,422,669
33.507
1020
2,220,819,602,560,918,840
45.028
1025
176,846,309,399,143,769,411,680
56.546
From studying these patterns so much, Gauss noticed an extraordinary connection between the number of
primes, and logarithms to base e, or the natural logarithm. This relationship comes from the fact that
2 ≈ 2.71823.... The first estimate Gauss found was denoted by π(x) ∼ logn n where π(x) is the number of primer
numbers between 1 and n. Although Gauss had found an effective way to see primes were indeed distributed
as randomly as previously thought, he could not find a way to prove his connection between prime numbers
and logarithms. He also found that as n got larger, the number of primes his formula predicted produced
results increasingly farther away than the actual number of primes.
Six years after, French mathematician, Legendre, created a similar formula for the same purpose, but Legendre’s
formula was obviously different from Gauss’s. In one way or another, Legendre’s formula had the opposite
condition as Gauss’s. As n increased, Legendre’s formula produced more and more accurate predictions of the
n
with B = −1.80366, which
number of primes. Legendre improved Gauss’s π(x) ∼ logn n to π(x) ∼ log(n)+B
is sometimes referred to as Legendre’s constant. It is said that although Legendre created his work six years
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after Gauss had made the same conjecture, Legendre was completely unaware that Gauss was working on the
same idea. After finding out that Legendre had created a formula more accurate than his, Gauss was set on
creating an even more accurate formula. He found that the probablility of getting a prime number up to n
can be seen as p(x) = log1 n where the sum of the primes is predicted to be log1 2 + log1 3 + ... + log1 n . From
here, Gauss showed that with a slight variation to the sum of probabilities, he had created what he called the
logarithmic integral,
R n which happened to be incredibly accurate. This new function is denoted by π(x) ∼ Li(n)
where Li(n) = 2 log1 x dx. By the time Gauss revealed his information to anyone other than himself, he had
created a table of primes up to 3,000,000. Although Legendre’s estimate seemed to become more accurate as
n became larger, this was only true for smaller values of n. This was proven by a Professor in the University
of Prague, who had created a list of primes up to 100,000,000 named Jakub Kulik. From these tables, Kulik
realized that Gauss’s estimate was much better than Legendre’s. The most surprising aspect of Gauss’s Li(n)
is the fact that it becomes more accurate as n becomes larger. This function would later become the prime
counting function in 1896, when Jacques Hadamard and Charles de la Vallée Poussin were able to prove that
Gauss’s prime conjecture was correctly using properties of the Riemann zeta function.
Bernhard Riemann is credited to having created one of the most influential formulas informing the distribution
of prime numbers, known as the Riemann zeta function. Using this function along with the Riemann Hypothesis, Riemann showed that primes have a behavior of asymptotic distribution, meaning they become more rare
as the numbers become larger. One of Riemann’s largest influences was his teacher Dirichlet. When working
with Dirichlet as his instructor, Riemann published a ten page paper on titled On the Number of Primes
Less Than a Given Magnitude. Even though this was the only paper Riemann published on number theory,
through its application, Riemann had created the most precise prime number function in history. While the
prime number theorem asserts that pn ≈ n ln(n), Riemann’s Hypothesis predicts a much more precise formula
√
for pn , where pn = |π(x) − Li(x) < 1/8π x log(x) for all x > 2657. What is more impressive is that the error
predicted by the Riemann Hypothesis is nearly the same as the error one would see if primes were randomly
distributed. This leads one to see that primes are pseudorandom, which means that although they appear
to behave randomly, they have a deterministic pattern. Here is a table comparing the accuracies of Gauss,
Legendre, and Riemann’s functions.
x
π(x)1
Gauss’s Li(x)
Legendre
Riemann
103
168
178
172
168.4
104
1,229
1,246
1,231
1,226.9
105
9,592
9,630
9,588
9,587.4
106
78,498
78,628
78,534
78,527.4
107
664,579
664,918
665,138
664,667.4
108
5,761,455
5,762,209
5,769,341
5,761,551.9
109
50,847,534
50,849,235
50,917,519
50,847,455.4
1010
455,052,511
455,055,614
455,743,004
455,050,683.3
Like most other people around his time, Riemann had a lot of difficulty proving his ideas. But unlike many
mathematicians, Riemann’s Hypothesis is still regarded as one of mathematics biggest mysteries of all time.
The reason this Hypothesis is so important is because if proved right, it would be able to locate all prime
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numbers in a given interval.
Today, prime numbers are not only a concern to mathematicians, it is also a concern to people to use the
internet, as prime numbers, and their ”randomness” is a critical component in computer security. At first,
even mathematicians thought that the study of primes was for a purely mathematical reason, and that they
would have no real life application. This was proven wrong when prime numbers were shown to even emerge
in nature as well as cryptography and computer security. There is also speculation that the zeros in the zeta
function are connected to energy levels in complex quantum physics. There is still a large amount of mystery
surrounding prime numbers, and it is sure to make many generations of mathematicians to come scratch their
heads in a wind of confusion.
Diving into the Past of Prime Numbers
As mentioned previously, prime numbers have been one of the biggest mysteries in the history of humanity.
While most think that the greeks were the first to make notice of prime numbers, this is only true because they
are the first to vastly begin to explore the numbers. In reality there is evidence that some ancient civilizations
much earlier than the greeks had knowledge of primes.
Figure 1: The Ishango Bone
One such pieces of evidence is the Ishango bone. The Ishango bone is the oldest known object to contain
logical carvings, it was discovered in the Congo, and it has been dated to be about 22,000 years old. The
bone contains carvings which many believe signify the implementation of a numerical system in the ancient
civilization. One of the columns of carvings contains all of the prime numbers between 10 and 20, being 11,
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13, 17, and 19. Unfortunately, no concrete connection can be made between this bone and the knowledge this
civilization had of prime numbers, because the prime quadruplet mentioned earlier (11, 13, 17, 19) is just a
partition of 60 into different odd numbers. This bone merely displays the earliest recorded group of prime
numbers discovered so far.
Figure 2: The Ishango Bone Carvings
Another piece of evidence is the work the Egyptians worked to develop. Mainly to support this is what is
known as an Egyptian Fraction. These were a way to express the rational numbers precicely, which would be
m
1
1
1
=
where a1 < a2 < ... < a`
+
+ ... +
n
a1 a2
a`
Through studying what is known as the Rhind papyrus, which has been used to be able to understand the
tables of expansions for numbers of the form n2 , it was shown that the expansions did not match any single
identity; instead, many of the different identities match the expansions of prime and composite denominators.
Although there is not much more evidence towards the knowledge of prime numbers previous to the greeks, it
is certain that primes have always been something of interest to anyone with interest in the world.
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Notation Worth Mentioning
Selberg’s Outline of the Prime Number Theorem makes use of a large amount of variables and notation that
anyone who unfamiliar with the mathematical language would have trouble understanding. Since this paper is
studying a specific section of that outline, I will only give the specifics of the variables and notations included.
Notation
log(x)= Logarithm/Natural Logarithm: A logarithm is the opposite of exponents, which would be x2 . This
means that they can be mathematically canceled out. The basic relationship between exponents and logarithms
shown with simple variables is y = bx ↔ logb (y). In this paper we will be using the logarithm to the base of
e, known as the natural logarithm, so unlike the logarithm just given, there will be no subscript because it is
assumed to be loge (x)
R(x) = θ(x) − x: with some simple substitution, this function can be rewritten as
R(x)
=0
x→+∞ x
lim
which will be used extensively in this section of the proof.
|x| = Absolute Value: The absolute value symbol is simply two lines around the quantity whose value is the
one you are looking for. In simple terms, the absolute value means the distance away from zero on the number
line. This means that |−10| = |10| = 10. The absolute value is used often in the outline due to the fact that
R(x) can be either positive or negative
P
= Sum: This symbol describes the sum of all quantities provided by the parameters of the symbol. The
data
P given under the symbol describes from where to when the sum should be accounted for. In the case of
log p The sum is being carried out for the value of p, from p being greater than y up to p being equal
y<p≤y 0
to or less than y’.
δ = Delta: This funny looking symbol is known as delta, and it is actually one of the letters of the greek
alphabet. In general in mathematics, delta is used as the difference between two given values, but in this
paper, delta is seen as a small number less than one.
y, y 0 = Number Variables: Since the Prime Number Theorem talks about a large amount of numbers an no
one specific number, variables such as y and y’ are used to denote different natural numbers. Throughout the
rest of the outline, other variables are used for this same purpose, such as n, x, and k.
p = Prime Number Variables: Similar to the Number Variables, the theorem doesn’t talk about one specific
prime number, so other variables are set aside specifically for prime numbers, as well as p, this includes r and
q.
O(x) = Landau Notation: This symbols, pronounced simply as ”o” of whatever is inside the parenthesis is
a symbol used to describe the growth of a given function, representing it in simplest terms. The O is used
because the rate of growth of the function is known as its order. A simple example is given the function
f (x) = 7x3 + 7x2 + 6x + 2, when we ignore constants and slower growing terms, we can say that f(x) grows at
an order of x3 , which can be rewritten to f (x) = O(x3 )
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Prime Number Theorem Outline
In this elementary exploration of the Prime Number Theorem (PNT) we are going to explain a specific section
of the PNT. The overall point of the PNT is to prove that
lim
x→∞
θ(x)
=1
x
where for all x > 0, θ(x) is defined as:
θ(x) =
X
log p
p≤x
with p defining all prime numbers.
From this we well jump over many steps in order to show a very compacted overview of the proof.
Step 1:
Proving Selber’s Formula:
X x
θ(x) log x +
log p = 2x log x + O(x)
θ
p
p≤x
Step 2:
After proving this formula, we derive the following inequality:
1 X x log log x
|R(x)| ≤
+O x
R
log x
n
log x
n≤x
Step 3:
In this step we must establish the existence of a constant K > 0 so that for any δ > 0 and any x > eK/δ , we
find that the interval x, xeK/δ contains a subinterval y, yeδ/2 so that every z in the interval yeilds:
|R(x)| < 4δz
Step 4:
Here, we show that if a < 8 is a positive number then the inequality |R(x)| < ax for x large enough, we arrive
at the new inequality:
a2
|R(x)| < a 1 −
x
300K
The step that we will be studying the most in this section is step 3.
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Section 3.3 of Selberg’s Proof
Directly taken out of Selberg’s paper, this is section 3.3 of the PNT Proof
|R(y)| < δy
from (2.10) we see that for y < y 0 ,
X
0≤
0
log p ≤ 2(y − y) + O
y<p≤y 0
from which follows that
R(y 0 ) − R(y) ≤ y 0 − y + O
y0
log y 0
y0
log y 0
,
.
Hence, if y/2 ≤ y 0 ≤ 2y, y > 4,
R(y 0 ) − R(y) ≤ y 0 − y + O
y0
log y 0
,
or
R(y 0 ) ≤ |R(y)| + y 0 − y + O
y0
log y 0
.
Break Down
To begin to break this down, we must subtract two fairly large equations to prove that y 0 > y > 0 for which
we have
log y 0
0
0
|R(y ) − R(y)| ≤ (y − y) + O
y0
From this, we replace y with y’ to get
0 X log p log q
y
0
log p +
= 2y + O
log(pq)
log y 0
0
0
X
p≤y
and
X
p≤y
pq≤y
X log p log q
y
log p +
= 2y + O
log(pq)
log y
pq≤y
We then subtract these two equalities to arrive to
X
log p +
y<p≤y 0
and, since the quantity
P
y<pq≤y 0
X
y<pq≤y 0
log p log q
log(pq)
log p log q
= 2y 0 − 2y + O
log(pq)
y0
log y 0
≥ 0 we can rewrite the whole equation to the inequality
X
log p ≤ 2y 0 − 2y + O
y<p≤y 0
8
y0
log y 0
or
X
0
log p ≤ 2(y − y) + O
y<p≤y 0
y0
log y 0
Then we must take the function given in the outline θ(x) and with some simple arithmetic, we show that
X
X
X
log p =
θ(y 0 ) − θ(y) =
log p −
log p
p≤y 0
From this, we replace
P
p≤y
y<p≤y 0
log p with θ(y 0 ) − θ(y) to arrive to
y<p≤y 0
0
0
0 ≤ θ(y ) − θ(y) ≤ 2(y − y) + O
y0
log y 0
At this point, we can do a few steps of arithmetic simplification to be able to get the value we want. Starting
with
0 y
0
0
0 ≤ θ(y ) − θ(y) ≤ 2(y − y) + O
log y 0
we subtract one of the two (y 0 − y) to get to
0
0
0
(θ(y ) − θ(y)) − (y − y) ≤ (y − y) + O
y0
log y 0
One might notice that the one on the left end of the inequality goes away. This is because the zero is needed
at the beginning to specify that the inequality is positive. once we rearrange the inequality, this is no longer
necessary.
After this, we rearrange the inequality to get
(θ(y 0 ) − y 0 ) − (θ(y) − y) ≤ (y 0 − y) + O
y0
log y 0
At this point, we can replace θ(x) − x with R(x) which we take from the notation section of the paper. With
this we get
0 y
0
0
R(y ) − R(y) ≤ (y − y) + O
log y 0
And finally, we add the absolute value signs around R(y 0 ) − R(y) to arrive to the inequality given in the proof,
which is
0 y
0
0
R(y ) − R(y) ≤ y − y + O
log y 0
We place the absolute value signs around R(y 0 ) − R(y) because if they become negative then the inequality is
no longer valid, so the absolute value signs make sure to keep the value positive.
From here, we want to have y’ be small enough, but not too small, so as it is said in the proof, if y/2 ≤ y 0 ≤
2y, y > 4, then we will need to place absolute value signs around (y 0 − y) to make sure that our value doesn’t
become negative. This is a simple change, simply rewriting the inequality to
0 y
R(y 0 ) − R(y) ≤ y 0 − y + O
log y 0
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At this point, we use the fact that
||a| − |b|| ≤ |a − b|
|a| − |b| ≤ |a − b|
|a| ≤ |b| + |a − b|
we rewrite the inequality to its final form in this section of the proof as
0 y
R(y 0 ) ≤ |R(y)| + y 0 − y + O
log y 0
This is our desired form of the inequality
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References
1. http://people.math.umass.edu/ tevelev/475 2014/laitinen.pdf
2. http://mathworld.wolfram.com/IshangoBone.html
3. https://en.wikipedia.org/wiki/Ishango bone#/media/File:Os %27Ishango IRSNB.JPG
4.http://www.storyofmathematics.com/egyptian.html
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