Download Elementary Number Theory Fall 2014 Lecture 4: Distribution of

Elementary Number Theory
Fall 2014
Lecture 4: Distribution of prime numbers
Distribution of prime numbers: There are infinitely many prime numbers. How are
these distributed among the natural numbers?
Define a function which gives the number of primes less than or equal to x.
π(x) = #{p ∈ P | p ≤ x}.
In 1792 Gausss conjectured that
π(x) ∼
x
as x 7→ ∞.
ln(x)
He later refined it to
∫
π(x) ∼ Li(x) =
2
x
dt
.
ln(t)
Hadamard and de la Vallee Poussin independently proved this conjecture in 1896. This
results is now known as the prime number theorem.
It follows that
π(x) ln(x)
= 1.
x−→∞
x
lim
1
Elementary Number Theory
Fall 2014
8
6
4
2
5
15
10
20
FIG. 1: π(x) for 1 ≤ x ≤ 20
25
20
15
10
5
20
40
60
80
100
FIG. 2: π(x) for 1 ≤ x ≤ 100
2
Elementary Number Theory
Fall 2014
150
125
100
75
50
25
200
400
600
800
1000
FIG. 3: π(x) for 1 ≤ x ≤ 1000
2200
2000
1800
1600
1400
12000 14000 16000 18000 20000
FIG. 4: π(x) for 10000 ≤ x ≤ 20000
3
Elementary Number Theory
Fall 2014
200
400
600
800
1000
8000
10000
1.18
1.16
1.14
FIG. 5:
π(x) ln(x)
2
for 2 ≤ x ≤ 1000
1.18
1.16
1.14
1.12
2000
FIG. 6:
4000
π(x) ln(x)
2
6000
for 2 ≤ x ≤ 10000
4
Elementary Number Theory
Fall 2014
1.14
1.13
1.12
1.11
20000 40000 60000 80000 100000
1.09
FIG. 7:
π(x) ln(x)
2
for 2 ≤ x ≤ 100000
1.07
1.065
1.06
1.055
10
2·10
10
4·10
10
6·10
10
8·10
11
1·10
1.045
FIG. 8:
π(x) ln(x)
2
for 2 ≤ x ≤ 1011
5
Elementary Number Theory
Fall 2014
4
3.5
3
2.5
2
1.5
4
FIG. 9: π(x) and
6
8
x
ln(x)
for 2 ≤ x ≤ 10
10
25
20
15
10
5
20
40
FIG. 10: π(x) and
60
x
ln(x)
80
100
for 2 ≤ x ≤ 100
6
Elementary Number Theory
Fall 2014
150
125
100
75
50
25
200
400
FIG. 11: π(x) and
600
x
ln(x)
800
1000
for 2 ≤ x ≤ 1000
1200
1000
800
600
400
200
2000
4000
FIG. 12: π(x) and
x
ln(x)
6000
8000
10000
for 2 ≤ x ≤ 10000
7