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.
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
R(x) =
|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 given under the symbol describes from where to when the sum should be accounted for. In the case of
P
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 )
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:
X
θ(x) =
logp
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:
θ(x)logx +
X x
θ
p≤x
p
logp = 2xlogx + O(x)
Step 2:
After proving this formula, we derive the following inequality:
|R(x)| ≤
x loglogx
1 X R
+O x
logx n≤x
n
logx
Step 3:
K
In this step we must establish
the existence of a constant
K > 0 so that for any δ > 0 and any x > e δ , we
K
δ
find that the interval x, xe δ contains a subinterval y, ye 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.
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 ,
0≤
X
log p ≤ 2(y 0 − y) + O
y<p≤y 0
2
y0
,
log y 0
from which follows that
R(y 0 ) − R(y) ≤ y 0 − y + O
y0
.
log y 0
Hence, if y/2 ≤ y 0 ≤ 2y, y > 4,
R(y 0 ) − R(y) ≤ y 0 − y + O
or
R(y 0 ) ≤ |R(y)| + y 0 − y + O
3
y0
,
log y 0
y0
.
log y 0