Prime Number Theorem
Dionna Bidny
Introduction
The series of prime numbers, i.e., those numbers > 1 solely divisible by themselves and 1, begins as follows:
2, 5, 7, 11, 13, 17, 19, 23, 29, ...,
The distribution of this series hold long-standing fascination to mathematicians, and can be generally approximated by the prime number theorem, the focus of this paper.
The prime number theorem, first proposed by Gauss in 1792, states that for π(x) = number of primes ≤ x
where x =a positive real number,
limx→∞ ( x/π(x)
log x ) = 1
This paper aims to introduce the historical events and thought processes that lead to a number of proofs of
the prime number theorem, as well as a discussion of portions the proof itself.
History
It is impossible to identify the point in history at which the concept of prime numbers first originated, but this
idea clearly dates far back into antiquity. Euclid, for example, outlined a number of statements about prime
numbers that formed the foundation to theorems surrounding this series for future generations. Among these
statements are the following:
1. (Fundamental Theorem of Arithmetic) Every positive integer n > 1 can be writted as a uniqe product of
one or more primes numbers
2. There exist an infinite number of primes.
An accurate distribution of these primes, however, is much more elusive then simply a definition of their
behavior. The first step in the deveopment of the theory of primes was performed by Gauss in 1792 − 3.
Previous accounts metnion that Gauss was so fascinated by the series of primes that he would spend many
hours writing out tables of the number of primes over every interval of 1000 to attempt to identify any trends
in their distribution. For example, the tables state that there are 168 prime numbers between 1 and 1000;
135 from 1001 to 2000, 127 from 3001 to 4000, and so on. Gauss continued to painstaikingly calculate prime
numbers far into the millions to attempt to demonstrate a distribution.
These calculations sparked a suspicion in Gauss. He had the idea that the density of primes in an interval
[a, b) around an integer n was 1/ log n, thus stating that the number of primes in that interval would be
Z b
dx
a
log x
To test this, Gauss created a secont set of tables that compared the number of primes found in each interval
with the answers to the above integral for each set. His results, which investigated primes up to 3.000, 000,
demonstrated a striking similarity between these two groups.
These tables and equations, while the first major step toward building the prime number theorem, were not
published until much later. The first published work to anticipate the prime number theorem is credited to
Legendre. Legendre first stated that π(x) has the form x/(A log x + B) where A and B are constants. He later
updated this statement and asserted the following:
π(x) =
x
log x + A(x)
where limx→∞ (A(x)) is ”approximately 1.08366...”
Legendre also asserted that there exist infinitely many primes of the form l + kn(n = 0, 1, 2, 3, ...) when k and
l are coprime numbers, ie, no positive integer besides 1 divides both k and l.
While an in-deapth description of their works is beyond the scope of this paper, Dirichlet and Tchebycheff too
offered major contributions to the devopment of the pirme numbertheorem. Dirchlet made incredible progress
in the realm of incorporating analytic methods into number theory. This offered a new angle with which to
view prime numbers and their analysis.
While Tchebycheff was not able to prove the theorem to completion, he did create another strong set of
fundamentals that led toward its proof. in 1848, he proved that the ratio of π(x) to x/ log x lies between
0.92 and 1.11 for sufficiently large x. This proof stemed from one of Tchebysheff’s equations for ϑ(x) which
P
defined ϑ(x) as p≤x log p. In summary, Tchebycheff showed that the prime number theorem is equal to
limx→∞ ( ϑ(x)
x ) = 1. We will further discuss the importance of Tchebycheff’s ϑ(x) equation and its role in the
fudamental proof of the prime number theorem in a later section. At its time, Tchebycheff’s proof became
the best proof for the prime number theorem,which was finally proved compeltely in 1896 independently by
Poussin and Hadamard.
The next and possibly most major step in the proof of the prime number theorem was the development of the
Riemann zeta function by Riemann in 1860. The Riemann zeta function is the infinite sum
ζ(s) =
∞
X
1
n=1
ns
For which the real part of s is greater than 1.
The proof developed by Poussin and Hadamard was not elementary and applied this function and Hadamard’s
theory of integral functions.
In 1948, Erdös and Sleberg announced that they devleoped a very elementary proof to the prime number
theorem which applied only very simple and fundamental logarithmic functions.
Eventually, however, this announcment and the continued involvment of the mathematicians in the prime
number theorem, ultimately led an unforunate dispute between them. The actual details of the dispute have
been blurred over time, but the premise is straightforward. In summary, both Erdös and Selberg used each
others’ work in their own developments of the proof for the prime number theorem. Thus, the question arose
whether a joint paper on the entire proof or rather two individual papers of their separate contributions ought
to be published. In a letter to Erdös, Selberg propsed the following:
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What I propose is the only fair thing: each of us can publish what he has actually done and get the credit for
that, and not for what the other has done.
You proved that
lim (
n→∞
pn+1
)=1
pn
I would never have dreamed of forcing you to write a joint paper on this in spite of the fact that the essential
thing in the proof of the result was mine.
Since there can be no reason for a joint paper, I am going to publish my proof as it now is. I have the opinion, .
. . that I do you full justice by telling in the paper that my original proof depended on your result. In addition
to this I offered you to withhold my proof so your theorem could be published earlier (of course then without
mentioning PNT).
I still offer you this. . .
If you dont accept this I publish my proof anyway.
Selberg went on to suggete that Erdös publish his results while he himself publishes his newly devloped proof
(partially based off of Erdös’ work) but with a refference to Erdös’ result. He offered to send the outline of his
proof to Erdös, stating that the prodigious journal Annals of Math could surely agree to publish Erdös’ work
before his own.
However, this angered Erdös, who reminded Selberg that Selberg had been doubtful of Erdös’ work and how
it implied the prime nuymber theorem. Erdós states that had he seen Selberg’s wok, Erdös would have been
able to use it to finish the proof of the prime number theorem then and there. If Erdös had kept his work
away from Selberg as Selberg did to him, Erdös would have succeeded in proving the prime number theorem
first and possibly alone. In addition, Erdös was adament against simply publishing his own work without
any relationship as to how it related to the prime number theorem. Thus, he agreed to send his paper to
Weyl for inspection, to ensure that it was thorough and inclusive but fair to Selberg when the two publish
separately.Ultimately, it was agreed that Selberg should publish his paper in Annals of Math and Erdös should
publish in the Bulletin. However, Jacobson at the Bulletin rejected Erdös’s paper. Soon after in a letter to
Jacobson, Weyl stated:
I had questioned whether Erdös has the right to publish things which are admittedly Selbergs...I really think
that Erdöss behavior is quite unreasonable, and if I were the responsible editor I think I would not be afraid of
rejecting his paper in this form. But there is another aspect of the matter. It is probably not as easy as Erdös
imagines to have his paper published in time in this country if the Bulletin rejects it. . . So it may be better
to let Erdös have his way. No great harm can be done by that. Selberg may feel offended and protest (and
that would be his right), but I am quite sure that the two papers Selbergs and Erdösss together will speak in
unmistakable language, and that the one who has really done harm to himself will be Erdös.
While the conflict between Erdös and Selberg was ultimately never quite subdued, it is evident that their
contributions together were most vital in creating what is known to be the most elementary proof of the prime
number theorem to date.
Proof Outline
In this section, we will present the detailed proof for an idea fundamental to the prime number theorem proof
and discuss its role in the bigger picture of the proof.
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Recall the prime number theorem:
lim (
x→∞
π(x)
)=1
x/ log x
Tchebycheff defined
ϑ(x) =
X
log p = 1
p≤x
According to Tchebycheff, the prime number theorem is equivalent to
lim (
x→∞
ϑ(x)
)=1
x
In this section, we will prove a statement that led to limx→∞ ( ϑ(x)
x ) = 1 which in turn laid the groundwork to
the prime number proof itself. The arguments presented in the following proof portion are similar throughout
the entirity of the PNT proof
We aim to prove ϑ(x) = O(x)
The ’Big O notation’ (called Landau’s symbol) states that there exists C : ϑ(x) ≤ Cx for x large enough. i.e.,
for an x sufficently close to ∞, limx→∞ ( ϑ(x)
x ) = C where C > 1. Intuitivley, this means that ϑ(x) does not
grow faster than x.
We begin by introducting the binomical coefficient. The binomial coefficent nk is the number of combinations
of k items from a set of size n. This value is also known as a comination or cominatorial number and is defined
by
n
k
!
=
n!
k!(n − k)!
Newton’s binomial formula reads as follows:
!
n
(a + b) =
!
!
!
n n
n n−1
n n−2 2
n n
a +
a
b+
a
b + ... +
b
0
1
2
n
Therefore, if we set a = 1, b = 1, n = 2n,
!
(1 + 1)
2n
2n
=2
=
!
!
2n
2n
2n
2n
+
+
+ ... +
0
1
2
2n
We see that
2n
n
!
< 22n
4
!
Also,
2n
n
=
(2n)!
n!n!
If p is a prime n < p ≤ 2n, then p divides the top but not the bottom. i.e., If p is a prime n < p ≤ 2n, then
2n
p|
n
!
Therefore,
2n
p|
n
n<p≤2n
!
Y
Y
2n
n
p≤
n<p≤2n
Y
!
< 22n
p < 22n
n<p≤2n
log(
Y
p) < log(22n )
n<p≤2n
By the law of logs:
X
log p < log(22n )
n<p≤2n
Because ϑ(2n) =
P
p≤2n log p
and ϑ(n) =
P
p≤n log p,
we get
ϑ(2n) − ϑ(n) < 2n log(2)
Define successively n = 2k−1 , 2k−1 , . . . , 2o = 1
Therefore if we sum the following inequalities:
ϑ(2k ) − ϑ(2k−1 ) < 2k log 2
ϑ(2k−1 ) − ϑ(2k−2 ) < 2k−1 log 2
.
.
.
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ϑ(2) − ϑ(1) < 2k log 2
Since 2(2k−1 ) = 2k and ϑ(1) = 0, we get:
ϑ(2k ) < 2(2k − 1) log 2 < 2(2k ) log 2
ϑ(2k ) < 2(2k ) log 2
For a fixed x, we have:
2k−1 ≤ x < 2k
ϑ(x) ≤ ϑ(2k ) < 2(2k ) log 2 = 4(2k−1 log 2 ≤ 4x log 2
which means that:
ϑ(x) < 4 log 2x
Therefore,
ϑ(x) = O(x)
Throught this, Tchebycheff was ultimatley able to prove the relationship between ϑ(x) and the prime number
theorem , thus presenting a major portion of the ground work for the prime number theorem proof itself.
Conclusion
Developing an understanding of the behavior of prime numbers has been the goal of mathematicians since
primes were first discovered. The development of the prime number theorem over the centuries exemplifies
our continually growing and expanding understanding of the distribution of these numbers. As potentially the
most useful and applicable whole numbers, primes apply to so many areas of mathematics and other fields,
and our understanding of them is of the most vital importance.
References
1. L. J. Goldstein, 1973, A History of the Prime Number Theorem, Pp.599 − 615 in The American Mathematical Monthly, Vol. 80
2. D. Zagier, 1997, Newman’s Short Proof of the Prime Number Theorem, Pp. 705 − 708 in The American
Mathematical Monthly, Vol. 104
3. D.Goldfeld, 2004 ’The Elementary Proof of the Prime Number Theorem: An Historical Perspective’ Pp.
179 − 192 in Number Theory, edited by D. Chudnovsky, G. Chudnovsky, M Nathanson. Springer New
York.
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