Selberg’s Prime Number Theorem Explained
Yassmin Saqri
Selberg’s life
A brilliant American mathematician known for his work in number theory, Atle Selberg astonished many in
1949 when he proved the prime number theorem. It all started on the 1917, as Anna Kristina Selberg and Ole
Michael Ludvigsen Selberg welcomed their baby boy Atle Selberg. Selberg was born in Langesund, Norway
and surely went on to becoming one of the most influential mathematicians of the twentieth century. As
Selberg was in school, it is easy to say that he was immensely influenced by the work of Srinivasa Ramanujan.
Specifically, Selberg found an exact analytical formula for the partition function as proposed by Ramanujan...
Erdös and Selberg Controversy
It all began in 1948 when the two collaborated to prove a basic proof that involved proving the prime number
theorem by using many logarithm manipulations. Once Selberg and Erdös finished the proof, it is said to be
that that is when the feuding started, however, it may have been distorted over time. It is considered that
Selberg had the idea of publishing separately which would ultimately discredit Erdös for all the work that he
has done. Of course Selberg said that he would give credit him, but this still did not sit well with Erdös. Some
say that Selberg even told Erdös not to mention the prime number theorem in his work at all.
Although many details have become blurred overtime, it is known that Erdös and Selberg continued to feud
that entailed Erdös to say that he could have kept the limit to himself and he would have produced the
prime number theorem alone. Erdös also claimed that without him, Selberg would have only be left with
his rudimentary inequality. All in all, as time went on, Selberg and Erdös never seemed to come to a firm
agreement in which Erdös believed he was receiving enough credit for his work...
History of the Prime Number Theorem
(add information of history when Selberg release PNT) (add info of other mathematicians)
Prerequisites
In order to completely understand what is going on in the prime number theorem, one must be familiar with
these topics:
1. What must first be noted is that a prime number is defined as:
A natural number greater than 1 that has no positive divisors other than 1 and itself.
2. ...
3.
4.
Outline of the Prime Number Theorem
Selberg’s proof rests on the formula:
θ(x)logx +
x
θ( )logp = 2xlogx + O(x)
p
p≤x
X
First of all, we need to recall that:
θ(x) =
X
logp
p≤x
We also need to remember that the Prime Number Theorem is equivalent to the statement:
lim
x→∞
θ(x)
=1
x
The formula that we first began with comes as a consequence of the Prime Number Theorem (PNT). One
might even describe it as a undeviating byproduct of the PNT. This is inferred through an averaging process
and in this case, the averaging process refers to the sum:
x
θ( )logp
p
p≤x
X
The task at hand is to show that Selberg’s formula implies the PNT.
... (continue from page one)
Explanation of part 17
We will first start out by writing the section of the proof that we will be breaking down. The section begins
by using the inequality when 4 ≤ x ≤ x0
|
X
x≤n≤x0
R(n)
| < K1
n2
We have, if R(n) does not change its sign within x and x’, then there exists a y in the interval x ≤ y ≤ x0 so
that,
|
R(y)
K2
|<
0 , K2 ≥ 1
y
log xx
This only holds true if R(n) changes the sign as well. This happens because then there will be a,
|R(y)| < log(y)
2
When an unknown fixed positive δ < 1and x > 4, there will exist a y in the interval x ≤ y ≤ e
|R(y)| < δy
Other releases of the PNT
(add a topic here)
References
3
K2
δ
, with