Download Prime Numbers and the Riemann Hypothesis

Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Prime Numbers and the Riemann Hypothesis
Benjamin Klopsch
Mathematisches Institut
Heinrich-Heine-Universität zu Düsseldorf
Tag der Forschung
◦
November 2005
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
“Investigation of the Distribution of Prime Numbers”
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What will we encounter in the next 45 minutes?
The ‘Building Blocks of Numbers’
What are prime numbers?
How many prime numbers are there?
Riemann – a Pioneer in Complex Analysis
Infinite Series and Complex Functions
Riemann and the Zeta Function ζ(s)
A ‘Wonderful Formula’
Riemann’s Formula
The Riemann Hypothesis
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What will we encounter in the next 45 minutes?
The ‘Building Blocks of Numbers’
What are prime numbers?
How many prime numbers are there?
Riemann – a Pioneer in Complex Analysis
Infinite Series and Complex Functions
Riemann and the Zeta Function ζ(s)
A ‘Wonderful Formula’
Riemann’s Formula
The Riemann Hypothesis
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What will we encounter in the next 45 minutes?
The ‘Building Blocks of Numbers’
What are prime numbers?
How many prime numbers are there?
Riemann – a Pioneer in Complex Analysis
Infinite Series and Complex Functions
Riemann and the Zeta Function ζ(s)
A ‘Wonderful Formula’
Riemann’s Formula
The Riemann Hypothesis
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001 = 7 · 143.
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001 = 7 · 11 · 13
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001 = 7 · 11 · 13
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
What are prime numbers?
Definition
A prime number is a natural number p admitting precisely two
distinct divisors, namely 1 and p.
Thus the first seven prime numbers are 2, 3, 5, 7, 11, 13, 17.
Theorem (Fundamental Theorem of Arithmetic)
Every natural number can written uniquely as a product of
prime numbers, up to re-ordering the factors.
For instance, the fairy tale number satisfies
1001 = 7 · 11 · 13
and similarly
112005 = 32 · 5 · 19 · 131.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
1
13
25
37
49
61
73
85
97
2
14
26
38
50
62
74
86
98
3
15
27
39
51
63
75
87
99
4
16
28
40
52
64
76
88
5
17
29
41
53
65
77
89
6
18
30
42
54
66
78
90
7
19
31
43
55
67
79
91
8
20
32
44
56
68
80
92
9
21
33
45
57
69
81
93
10
22
34
46
58
70
82
94
11
23
35
47
59
71
83
95
12
24
36
48
60
72
84
96
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
1
13
25
37
49
61
73
85
97
2
14
26
38
50
62
74
86
98
3
15
27
39
51
63
75
87
99
4
16
28
40
52
64
76
88
5
17
29
41
53
65
77
89
6
18
30
42
54
66
78
90
7
19
31
43
55
67
79
91
8
20
32
44
56
68
80
92
9
21
33
45
57
69
81
93
10
22
34
46
58
70
82
94
11
23
35
47
59
71
83
95
12
24
36
48
60
72
84
96
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
13
25
37
49
61
73
85
97
2
14
26
38
50
62
74
86
98
3
15
27
39
51
63
75
87
99
4
16
28
40
52
64
76
88
5
17
29
41
53
65
77
89
6
18
30
42
54
66
78
90
7
19
31
43
55
67
79
91
8
20
32
44
56
68
80
92
9
21
33
45
57
69
81
93
10
22
34
46
58
70
82
94
11
23
35
47
59
71
83
95
12
24
36
48
60
72
84
96
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
13
25
37
49
61
73
85
97
2
14
26
38
50
62
74
86
98
3
15
27
39
51
63
75
87
99
4
16
28
40
52
64
76
88
5
17
29
41
53
65
77
89
6
18
30
42
54
66
78
90
7
19
31
43
55
67
79
91
8
20
32
44
56
68
80
92
9
21
33
45
57
69
81
93
10
22
34
46
58
70
82
94
11
23
35
47
59
71
83
95
12
24
36
48
60
72
84
96
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
25
37
49
61
73
85
97
3
15
27
39
51
63
75
87
99
5
17
29
41
53
65
77
89
7
19
31
43
55
67
79
91
9
21
33
45
57
69
81
93
11
23
35
47
59
71
83
95
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
25
37
49
61
73
85
97
3
15
27
39
51
63
75
87
99
5
17
29
41
53
65
77
89
7
19
31
43
55
67
79
91
9
21
33
45
57
69
81
93
11
23
35
47
59
71
83
95
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
25
37
49
61
73
85
97
3
15
27
39
51
63
75
87
99
5
17
29
41
53
65
77
89
7
19
31
43
55
67
79
91
9
21
33
45
57
69
81
93
11
23
35
47
59
71
83
95
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
25
37
49
61
73
85
97
3
5
17
29
41
53
65
77
89
7
19
31
43
55
67
79
91
11
23
35
47
59
71
83
95
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
25
37
49
61
73
85
97
3
5
17
29
41
53
65
77
89
7
19
31
43
55
67
79
91
11
23
35
47
59
71
83
95
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
37
49
61
73
97
3
5
17
29
41
53
77
89
7
19
31
43
67
79
91
11
23
47
59
71
83
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
37
49
61
73
97
3
5
17
29
41
53
77
89
7
19
31
43
67
79
91
11
23
47
59
71
83
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
37
3
5
17
29
41
53
61
73
67
79
89
97
7
19
31
43
11
23
47
59
71
83
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Sieve of Erathostenes
Among the first 100 numbers there are 25 prime numbers.
2
13
37
3
5
17
29
41
53
61
73
67
79
89
97
7
19
31
43
11
23
47
59
71
83
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
There are infinitely many prime numbers
Theorem (Euclid, around 300 BC)
There are infinitely many prime numbers.
Proof.
• Assume, p1 = 2, p2 = 3, . . . , pr are all the prime numbers.
• Set N := p1 · p2 · · · pr + 1.
• Then N ≥ 2, but N is not divisible by any prime number.
• Contradiction!
Question: How many is “infinitely many”?
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
There are infinitely many prime numbers
Theorem (Euclid, around 300 BC)
There are infinitely many prime numbers.
Proof.
• Assume, p1 = 2, p2 = 3, . . . , pr are all the prime numbers.
• Set N := p1 · p2 · · · pr + 1.
• Then N ≥ 2, but N is not divisible by any prime number.
• Contradiction!
Question: How many is “infinitely many”?
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
There are infinitely many prime numbers
Theorem (Euclid, around 300 BC)
There are infinitely many prime numbers.
Proof.
• Assume, p1 = 2, p2 = 3, . . . , pr are all the prime numbers.
• Set N := p1 · p2 · · · pr + 1.
• Then N ≥ 2, but N is not divisible by any prime number.
• Contradiction!
Question: How many is “infinitely many”?
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
There are infinitely many prime numbers
Theorem (Euclid, around 300 BC)
There are infinitely many prime numbers.
Proof.
• Assume, p1 = 2, p2 = 3, . . . , pr are all the prime numbers.
• Set N := p1 · p2 · · · pr + 1.
• Then N ≥ 2, but N is not divisible by any prime number.
• Contradiction!
Question: How many is “infinitely many”?
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
There are infinitely many prime numbers
Theorem (Euclid, around 300 BC)
There are infinitely many prime numbers.
Proof.
• Assume, p1 = 2, p2 = 3, . . . , pr are all the prime numbers.
• Set N := p1 · p2 · · · pr + 1.
• Then N ≥ 2, but N is not divisible by any prime number.
• Contradiction!
Question: How many is “infinitely many”?
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
There are infinitely many prime numbers
Theorem (Euclid, around 300 BC)
There are infinitely many prime numbers.
Proof.
• Assume, p1 = 2, p2 = 3, . . . , pr are all the prime numbers.
• Set N := p1 · p2 · · · pr + 1.
• Then N ≥ 2, but N is not divisible by any prime number.
• Contradiction!
Question: How many is “infinitely many”?
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Number of Primes up to a Given Size
Definition
For every real number x let π(x) denote the
number of primes between 1 and x.
x
100
1000
10, 000
100, 000
1, 000, 000
10, 000, 000
100, 000, 000
π(x)
25
168
1, 229
9, 592
78, 498
664, 579
5, 761, 455
x/π(x)
4.0
≈ 6.0
≈ 8.1
≈ 10.4
≈ 12.7
≈ 15.0
≈ 17.4
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Number of Primes up to a Given Size
Definition
For every real number x let π(x) denote the
number of primes between 1 and x.
x
100
1000
10, 000
100, 000
1, 000, 000
10, 000, 000
100, 000, 000
π(x)
25
168
1, 229
9, 592
78, 498
664, 579
5, 761, 455
x/π(x)
4.0
≈ 6.0
≈ 8.1
≈ 10.4
≈ 12.7
≈ 15.0
≈ 17.4
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Number of Primes up to a Given Size
Definition
For every real number x let π(x) denote the
number of primes between 1 and x.
x
100
1000
10, 000
100, 000
1, 000, 000
10, 000, 000
100, 000, 000
π(x)
25
168
1, 229
9, 592
78, 498
664, 579
5, 761, 455
x/π(x)
4.0
≈ 6.0
≈ 8.1
≈ 10.4
≈ 12.7
≈ 15.0
≈ 17.4
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
Observation: Viewed from a distance, the function π(x) is
surprisingly smooth. The ‘density’ of primes around a large
number x is about 1/log(x).
Remark: log(x) ≈ 2.3 · (# digits before the decimal point of x).
Thus one has approximately π(x) ≈ x/ log(x).
Gauß suggested the following logarithmic integral Li(x) as a
more precise approximation:
Z x
1
1
1
1
dt ≈
+
+ ... +
.
Li(x) :=
log(2) log(3)
logbxc
2 log(t)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
Observation: Viewed from a distance, the function π(x) is
surprisingly smooth. The ‘density’ of primes around a large
number x is about 1/log(x).
Remark: log(x) ≈ 2.3 · (# digits before the decimal point of x).
Thus one has approximately π(x) ≈ x/ log(x).
Gauß suggested the following logarithmic integral Li(x) as a
more precise approximation:
Z x
1
1
1
1
dt ≈
+
+ ... +
.
Li(x) :=
log(2) log(3)
logbxc
2 log(t)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
Observation: Viewed from a distance, the function π(x) is
surprisingly smooth. The ‘density’ of primes around a large
number x is about 1/log(x).
Remark: log(x) ≈ 2.3 · (# digits before the decimal point of x).
Thus one has approximately π(x) ≈ x/ log(x).
Gauß suggested the following logarithmic integral Li(x) as a
more precise approximation:
Z x
1
1
1
1
Li(x) :=
dt ≈
+
+ ... +
.
log(2) log(3)
logbxc
2 log(t)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
Observation: Viewed from a distance, the function π(x) is
surprisingly smooth. The ‘density’ of primes around a large
number x is about 1/log(x).
Remark: log(x) ≈ 2.3 · (# digits before the decimal point of x).
Thus one has approximately π(x) ≈ x/ log(x).
Gauß suggested the following logarithmic integral Li(x) as a
more precise approximation:
Z x
1
1
1
1
Li(x) :=
dt ≈
+
+ ... +
.
log(2) log(3)
logbxc
2 log(t)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
Observation: Viewed from a distance, the function π(x) is
surprisingly smooth. The ‘density’ of primes around a large
number x is about 1/log(x).
Remark: log(x) ≈ 2.3 · (# digits before the decimal point of x).
Thus one has approximately π(x) ≈ x/ log(x).
Gauß suggested the following logarithmic integral Li(x) as a
more precise approximation:
Z x
1
1
1
1
Li(x) :=
dt ≈
+
+ ... +
.
log(2) log(3)
logbxc
2 log(t)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Logarithmic Integral Function
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
We build infinite series
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
• The harmonic series diverges to infinity:
∞
X
1
n=1
n
=
1+
1 1 1 1
+ + + + ...
2 3 4 5
=
∞.
to the proof
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
For
∞
X
1
n=1
1
=1+ +
n
2
1 1
+
3 4
+
1 1 1 1
+ + +
5 6 7 8
+ ...
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
For
∞
X
1
n=1
1
=1+ +
n
2
1
≥1+ +
2
1 1
+
3 4
1 1
+
4 4
+
+
1 1 1 1
+ + +
5 6 7 8
1 1 1 1
+ + +
8 8 8 8
+ ...
+ ...
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
For
∞
X
1
n=1
1
=1+ +
n
2
1 1
+
3 4
+
1 1 1 1
+ + +
5 6 7 8
+ ...
1 1
1 1 1 1
1
+
+
+ + +
+ ...
≥1+ +
2
4 4
8 8 8 8
1 2 4
= 1 + + + + ...
2 4 8
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
For
∞
X
1
n=1
1
=1+ +
n
2
1 1
+
3 4
+
1 1 1 1
+ + +
5 6 7 8
+ ...
1 1
1 1 1 1
1
+
+
+ + +
+ ...
≥1+ +
2
4 4
8 8 8 8
1 2 4
= 1 + + + + ...
2 4 8
1 1 1
= 1 + + + + . . . = ∞.
2 2 2
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
• The harmonic series diverges to infinity:
∞
X
1
n=1
n
=
1+
1 1 1 1
+ + + + ...
2 3 4 5
=
∞.
to the proof
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
• The harmonic series diverges to infinity:
∞
X
1
n=1
=
n
1+
1 1 1 1
+ + + + ...
2 3 4 5
=
∞.
• In contrast, the sum of quadratic reciprocals
∞
X
1
n2
n=1
=
1+
1
1
+ 2 + ...
2
2
3
is bounded above and converges.
=
1+
1 1
+ + ...
4 9
to the proof
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
We build infinite series
For
∞
∞
X
X
1
1
=
1
+
2
n
n2
n=1
n=2
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
We build infinite series
For
∞
∞
X
X
1
1
=
1
+
2
n
n2
n=1
n=2
≤
1+
∞
X
n=2
1
(n − 1)n
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
For
∞
∞
X
X
1
1
=
1
+
2
n
n2
n=1
=1+
n=2
∞
X
n=2
≤
1+
n − (n − 1)
(n − 1)n
∞
X
n=2
=
1
(n − 1)n
1+
∞ X
n=2
1
1
−
n−1 n
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
We build infinite series
For
∞
∞
X
X
1
1
=
1
+
2
n
n2
n=1
n=2
∞
X
≤
1+
∞
X
n=2
1
(n − 1)n
∞ X
n − (n − 1)
1
1
=1+
= 1+
−
(n − 1)n
n−1 n
n=2
n=2
1
1 1
1 1
=1+ 1−
+
−
+
−
+ ...
2
2 3
3 4
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
We build infinite series
For
∞
∞
X
X
1
1
=
1
+
2
n
n2
n=1
n=2
∞
X
≤
1+
∞
X
n=2
1
(n − 1)n
∞ X
n − (n − 1)
1
1
=1+
= 1+
−
(n − 1)n
n−1 n
n=2
n=2
1
1 1
1 1
=1+ 1−
+
−
+
−
+ ...
2
2 3
3 4
1 1
1 1
1 1
+ − +
+ − +
+ ...
=1+1+ − +
2 2
3 3
4 4
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
We build infinite series
For
∞
∞
X
X
1
1
=
1
+
2
n
n2
n=1
n=2
∞
X
≤
1+
∞
X
n=2
1
(n − 1)n
∞ X
n − (n − 1)
1
1
=1+
= 1+
−
(n − 1)n
n−1 n
n=2
n=2
1
1 1
1 1
=1+ 1−
+
−
+
−
+ ...
2
2 3
3 4
1 1
1 1
1 1
+ − +
+ − +
+ ...
=1+1+ − +
2 2
3 3
4 4
= 1 + 1 + 0 + 0 + 0 + . . . = 2.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
• The harmonic series diverges to infinity:
∞
X
1
n=1
=
n
1+
1 1 1 1
+ + + + ...
2 3 4 5
=
∞.
• In contrast, the sum of quadratic reciprocals
∞
X
1
n2
n=1
=
1+
1
1
+ 2 + ...
2
2
3
is bounded above and converges.
=
1+
1 1
+ + ...
4 9
to the proof
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
We build infinite series
• The harmonic series diverges to infinity:
∞
X
1
n=1
=
n
1+
1 1 1 1
+ + + + ...
2 3 4 5
=
∞.
• In contrast, the sum of quadratic reciprocals
∞
X
1
n2
n=1
=
1+
1
1
+ 2 + ...
2
2
3
=
1+
1 1
+ + ...
4 9
is bounded above and converges.
Question: What is the sum of all the quadratic reciprocals?
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Euler and the ‘Real Zeta Function’
One has
∞
X
1
π2
=
≈ 1.6449,
6
n2
n=1
where π = 3.1415....
Euler considered more generally the real function
∞
X
1
ζ(s) =
ns
for s > 1
n=1
and he computed ζ(2m) for all even numbers 2, 4,
6, 8, . . .
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Euler and the ‘Real Zeta Function’
One has
∞
X
1
π2
=
≈ 1.6449,
6
n2
n=1
where π = 3.1415....
Euler considered more generally the real function
∞
X
1
ζ(s) =
ns
for s > 1
n=1
and he computed ζ(2m) for all even numbers 2, 4,
6, 8, . . .
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Euler and the ‘Real Zeta Function’
One has
∞
X
1
π2
=
≈ 1.6449,
6
n2
n=1
where π = 3.1415....
Euler considered more generally the real function
∞
X
1
ζ(s) =
ns
for s > 1
n=1
and he computed ζ(2m) for all even numbers 2, 4,
6, 8, . . .
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Euler Product Formula
The Euler Product Formula yields a interesting connection
between the function ζ(s) and prime numbers:
1
1
1
1
1
ζ(s) = 1 + s + s + s + s + s + . . .
3
4
5
6 2
1
1
1
1
1
1
= 1 + s + 2s + 3s + . . . · 1 + s + 2s + 3s + . . .
2
3
2
2
3
3
1
1
1
· 1 + s + 2s + 3s + . . . · · ·
5
5
5
Y 1
1
1
=
1 + s + 2s + 3s + . . .
p
p
p
p prime
=
Y
p prime
1
.
1 − p−s
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Complex Number Plane
imaginäre Achse
2i
The familiar real number line R
extends to the so-called
complex number plane
C = R + iR.
i
−1
1
2
3
reelle Achse
−i
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
−i
The complex nmber i, called the imaginary unit, has the
property that its square equals −1.
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
imaginäre Achse
z = 2.5 + 2 i
2i
z+w=
i
−1
3.5 + 1.5 i
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
The basic arithmetic operations
addition, subtraction,
multiplication and division
extend from the real numbers
to the complex numbers.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Complex Number Plane
imaginäre Achse
2i
The familiar real number line R
extends to the so-called
complex number plane
C = R + iR.
i
−1
1
2
3
reelle Achse
−i
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
−i
The complex nmber i, called the imaginary unit, has the
property that its square equals −1.
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
imaginäre Achse
z = 2.5 + 2 i
2i
z+w=
i
−1
3.5 + 1.5 i
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
The basic arithmetic operations
addition, subtraction,
multiplication and division
extend from the real numbers
to the complex numbers.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Complex Number Plane
imaginäre Achse
2i
The familiar real number line R
extends to the so-called
complex number plane
C = R + iR.
i
−1
1
2
3
reelle Achse
−i
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
−i
The complex nmber i, called the imaginary unit, has the
property that its square equals −1.
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
imaginäre Achse
z = 2.5 + 2 i
2i
z+w=
i
−1
3.5 + 1.5 i
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
The basic arithmetic operations
addition, subtraction,
multiplication and division
extend from the real numbers
to the complex numbers.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Complex Functions
Simple transformations of the complex number plane to itself,
such as the polynomial function
f (z) = f (x + iy ) := z 3 − 64z,
can be visualised as landscapes.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Complex Functions
Simple transformations of the complex number plane to itself,
such as the polynomial function
f (z) = f (x + iy ) := z 3 − 64z,
can be visualised as landscapes.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Georg Friedrich Bernhard Riemann
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Georg Friedrich Bernhard Riemann
• born 1826 in Breselenz, Lüchow-Dannenberg
• student in Göttingen (initially in Theology) and Berlin
• 1851 PhD with the seminal Inauguraldissertation
“Grundlagen für eine allgemeine Theorie der Functionen
einer veränderlichen complexen Grösse”
• 1854 Habilitation lecture “Über die Hypothesen, welche der
Geometrie zu Grunde liegen” - a cornerstone of modern
Differential Geometry
• 1859 successor of Dirichlet to the Gauß-Chair, publication
of the article “Ueber die Anzahl der Primzahlen unter einer
gegebenen Grösse”
• deceased 1866 from tuberculosis in Selasca, Lago
Maggiore, Italy
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Georg Friedrich Bernhard Riemann
• born 1826 in Breselenz, Lüchow-Dannenberg
• student in Göttingen (initially in Theology) and Berlin
• 1851 PhD with the seminal Inauguraldissertation
“Grundlagen für eine allgemeine Theorie der Functionen
einer veränderlichen complexen Grösse”
• 1854 Habilitation lecture “Über die Hypothesen, welche der
Geometrie zu Grunde liegen” - a cornerstone of modern
Differential Geometry
• 1859 successor of Dirichlet to the Gauß-Chair, publication
of the article “Ueber die Anzahl der Primzahlen unter einer
gegebenen Grösse”
• deceased 1866 from tuberculosis in Selasca, Lago
Maggiore, Italy
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Georg Friedrich Bernhard Riemann
• born 1826 in Breselenz, Lüchow-Dannenberg
• student in Göttingen (initially in Theology) and Berlin
• 1851 PhD with the seminal Inauguraldissertation
“Grundlagen für eine allgemeine Theorie der Functionen
einer veränderlichen complexen Grösse”
• 1854 Habilitation lecture “Über die Hypothesen, welche der
Geometrie zu Grunde liegen” - a cornerstone of modern
Differential Geometry
• 1859 successor of Dirichlet to the Gauß-Chair, publication
of the article “Ueber die Anzahl der Primzahlen unter einer
gegebenen Grösse”
• deceased 1866 from tuberculosis in Selasca, Lago
Maggiore, Italy
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Georg Friedrich Bernhard Riemann
• born 1826 in Breselenz, Lüchow-Dannenberg
• student in Göttingen (initially in Theology) and Berlin
• 1851 PhD with the seminal Inauguraldissertation
“Grundlagen für eine allgemeine Theorie der Functionen
einer veränderlichen complexen Grösse”
• 1854 Habilitation lecture “Über die Hypothesen, welche der
Geometrie zu Grunde liegen” - a cornerstone of modern
Differential Geometry
• 1859 successor of Dirichlet to the Gauß-Chair, publication
of the article “Ueber die Anzahl der Primzahlen unter einer
gegebenen Grösse”
• deceased 1866 from tuberculosis in Selasca, Lago
Maggiore, Italy
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Georg Friedrich Bernhard Riemann
• born 1826 in Breselenz, Lüchow-Dannenberg
• student in Göttingen (initially in Theology) and Berlin
• 1851 PhD with the seminal Inauguraldissertation
“Grundlagen für eine allgemeine Theorie der Functionen
einer veränderlichen complexen Grösse”
• 1854 Habilitation lecture “Über die Hypothesen, welche der
Geometrie zu Grunde liegen” - a cornerstone of modern
Differential Geometry
• 1859 successor of Dirichlet to the Gauß-Chair, publication
of the article “Ueber die Anzahl der Primzahlen unter einer
gegebenen Grösse”
• deceased 1866 from tuberculosis in Selasca, Lago
Maggiore, Italy
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Georg Friedrich Bernhard Riemann
• born 1826 in Breselenz, Lüchow-Dannenberg
• student in Göttingen (initially in Theology) and Berlin
• 1851 PhD with the seminal Inauguraldissertation
“Grundlagen für eine allgemeine Theorie der Functionen
einer veränderlichen complexen Grösse”
• 1854 Habilitation lecture “Über die Hypothesen, welche der
Geometrie zu Grunde liegen” - a cornerstone of modern
Differential Geometry
• 1859 successor of Dirichlet to the Gauß-Chair, publication
of the article “Ueber die Anzahl der Primzahlen unter einer
gegebenen Grösse”
• deceased 1866 from tuberculosis in Selasca, Lago
Maggiore, Italy
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Georg Friedrich Bernhard Riemann
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Zeta Function
The infinite series
ζ(s)
=
∞
X
1
ns
n=1
=
1+
1
1
1
+ s + s + ...
s
2
3
4
converges for all complex numbers s = x + yi with real part
x > 1.
Moreover, the function ζ(s), extends uniquely to the entire
complex number plane, apart from a pole at s = 1.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Zeta Function
The infinite series
ζ(s)
=
∞
X
1
ns
n=1
=
1+
1
1
1
+ s + s + ...
s
2
3
4
converges for all complex numbers s = x + yi with real part
x > 1.
Moreover, the function ζ(s), extends uniquely to the entire
complex number plane, apart from a pole at s = 1.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Zeta Function
The infinite series
ζ(s)
=
∞
X
1
ns
n=1
=
1+
1
1
1
+ s + s + ...
s
2
3
4
converges for all complex numbers s = x + yi with real part
x > 1.
Moreover, the function ζ(s), extends uniquely to the entire
complex number plane, apart from a pole at s = 1.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Critical Strip
The ‘real’ function ζ(s) can be extended to the
entire complex number plane by means of the
Functional Equation
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
−10 i
−20 i
imaginäre Achse
Λ(s) := π −s/2 Γ(s/2)ζ(s) = Λ(1 − s).
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
−20 i
imaginäre Achse
30 i
...
kritischer Streifen
20 i
0.5 + i (21.02...)
0.5 + i (14.13...)
10 i
nicht−triviale
Nullstellen
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
Of particular importance are the zeros of Λ(s).
They correspond to the non-trivial zeros of ζ(s)
and lie in the critical strip
S := {x + yi | 0 ≤ x ≤ 1}.
0.5 − i (14.13...)
−20 i
imaginäre Achse
0.5 − i (21.02...)
...
All known zeros of Λ(s) lie on the line
G := {x + yi | x = 1/2}.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Critical Strip
The ‘real’ function ζ(s) can be extended to the
entire complex number plane by means of the
Functional Equation
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
−10 i
−20 i
imaginäre Achse
Λ(s) := π −s/2 Γ(s/2)ζ(s) = Λ(1 − s).
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
−20 i
imaginäre Achse
30 i
...
kritischer Streifen
20 i
0.5 + i (21.02...)
0.5 + i (14.13...)
10 i
nicht−triviale
Nullstellen
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
Of particular importance are the zeros of Λ(s).
They correspond to the non-trivial zeros of ζ(s)
and lie in the critical strip
S := {x + yi | 0 ≤ x ≤ 1}.
0.5 − i (14.13...)
−20 i
imaginäre Achse
0.5 − i (21.02...)
...
All known zeros of Λ(s) lie on the line
G := {x + yi | x = 1/2}.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Critical Strip
The ‘real’ function ζ(s) can be extended to the
entire complex number plane by means of the
Functional Equation
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
−10 i
−20 i
imaginäre Achse
Λ(s) := π −s/2 Γ(s/2)ζ(s) = Λ(1 − s).
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
−20 i
imaginäre Achse
30 i
...
kritischer Streifen
20 i
0.5 + i (21.02...)
0.5 + i (14.13...)
10 i
nicht−triviale
Nullstellen
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
Of particular importance are the zeros of Λ(s).
They correspond to the non-trivial zeros of ζ(s)
and lie in the critical strip
S := {x + yi | 0 ≤ x ≤ 1}.
0.5 − i (14.13...)
−20 i
imaginäre Achse
0.5 − i (21.02...)
...
All known zeros of Λ(s) lie on the line
G := {x + yi | x = 1/2}.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Chebyshev ψ-Function
Recall: We are to show that, for large x, the number π(x) of
prime numbers less than or equal to x is closely approximated
Grafik
by the logarithmic integral Li(x).
More handy thanPthe prime counting
function π(x) = p≤x 1 is a variant,
introduced by Chebyshev,
X
ψ(x) :=
log(p),
pk ≤x
which counts prime numbers with a
logarithmic weight.
Loosely speaking, the approximation π(x) ≈ Li(x)
is equivalent to ψ(x) ≈ x.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Chebyshev ψ-Function
Recall: We are to show that, for large x, the number π(x) of
prime numbers less than or equal to x is closely approximated
Grafik
by the logarithmic integral Li(x).
More handy thanPthe prime counting
function π(x) = p≤x 1 is a variant,
introduced by Chebyshev,
X
ψ(x) :=
log(p),
pk ≤x
which counts prime numbers with a
logarithmic weight.
Loosely speaking, the approximation π(x) ≈ Li(x)
is equivalent to ψ(x) ≈ x.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Chebyshev ψ-Function
Recall: We are to show that, for large x, the number π(x) of
prime numbers less than or equal to x is closely approximated
Grafik
by the logarithmic integral Li(x).
More handy thanPthe prime counting
function π(x) = p≤x 1 is a variant,
introduced by Chebyshev,
X
ψ(x) :=
log(p),
pk ≤x
which counts prime numbers with a
logarithmic weight.
Loosely speaking, the approximation π(x) ≈ Li(x)
is equivalent to ψ(x) ≈ x.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Chebyshev ψ-Function
Recall: We are to show that, for large x, the number π(x) of
prime numbers less than or equal to x is closely approximated
Grafik
by the logarithmic integral Li(x).
More handy thanPthe prime counting
function π(x) = p≤x 1 is a variant,
introduced by Chebyshev,
X
ψ(x) :=
log(p),
pk ≤x
which counts prime numbers with a
logarithmic weight.
Loosely speaking, the approximation π(x) ≈ Li(x)
is equivalent to ψ(x) ≈ x.
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Riemann’s Formula for the Prime Counting Function
The step function ψ(x) can be expressed precisely in terms of
the complex zeros of the Riemann zeta function:
X xρ
1
1
ψ· (x) = x − log(2π) −
− log 1 − 2 .
ρ
2
x
ρ
The following graphics visualise how the prime counting
function ψ(x) is stepwise approximated using the first 300 pairs
Graphics
of zeros of the Riemann zeta function.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Riemann’s Formula for the Prime Counting Function
The step function ψ(x) can be expressed precisely in terms of
the complex zeros of the Riemann zeta function:
X xρ
1
1
ψ· (x) = x − log(2π) −
− log 1 − 2 .
ρ
2
x
ρ
The following graphics visualise how the prime counting
function ψ(x) is stepwise approximated using the first 300 pairs
Graphics
of zeros of the Riemann zeta function.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Riemann’s Formula for the Prime Counting Function
The step function ψ(x) can be expressed precisely in terms of
the complex zeros of the Riemann zeta function:
X xρ
1
1
ψ· (x) = x − log(2π) −
− log 1 − 2 .
ρ
2
x
ρ
The following graphics visualise how the prime counting
function ψ(x) is stepwise approximated using the first 300 pairs
Graphics
of zeros of the Riemann zeta function.
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Stepwise Approximation of ψ(x)
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Hypothesis
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function lie on the line
G = {x + yi | x = 1/2}.
To this day, Riemann’s conjecture has not been proved or
disproved.
The Riemann Hypothesis is equivalent to the following
assertion.
Equivalent Formulation of the Riemann Hypothesis
There is a constant C such
√ that π(x) does not deviate from
Li(x) by more than C · x log(x).
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Hypothesis
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function lie on the line
G = {x + yi | x = 1/2}.
To this day, Riemann’s conjecture has not been proved or
disproved.
The Riemann Hypothesis is equivalent to the following
assertion.
Equivalent Formulation of the Riemann Hypothesis
There is a constant C such
√ that π(x) does not deviate from
Li(x) by more than C · x log(x).
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Hypothesis
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function lie on the line
G = {x + yi | x = 1/2}.
To this day, Riemann’s conjecture has not been proved or
disproved.
The Riemann Hypothesis is equivalent to the following
assertion.
Equivalent Formulation of the Riemann Hypothesis
There is a constant C such
√ that π(x) does not deviate from
Li(x) by more than C · x log(x).
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Hypothesis
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function lie on the line
G = {x + yi | x = 1/2}.
To this day, Riemann’s conjecture has not been proved or
disproved.
The Riemann Hypothesis is equivalent to the following
assertion.
Equivalent Formulation of the Riemann Hypothesis
There is a constant C such
√ that π(x) does not deviate from
Li(x) by more than C · x log(x).
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The Riemann Hypothesis
Riemann Hypothesis
All non-trivial zeros of the Riemann zeta function lie on the line
G = {x + yi | x = 1/2}.
To this day, Riemann’s conjecture has not been proved or
disproved.
The Riemann Hypothesis is equivalent to the following
assertion.
Equivalent Formulation of the Riemann Hypothesis
There is a constant C such
√ that π(x) does not deviate from
Li(x) by more than C · x log(x).
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
The Riemann Hypothesis
"‘. . . Man findet nun in der That etwa so viel reelle
Wurzeln innerhalb dieser Grenzen, und es ist sehr
wahrscheinlich, dass alle Wurzeln reell sind. Hiervon
wäre allerdings ein strenger Beweis zu wünschen; ich
habe indess die Aufsuchung desselben nach einigen
flüchtigen vergeblichen Versuchen vorläufig bei Seite
gelassen, da er für den nächsten Zweck meiner
Untersuchung entbehrlich schien. . . . "’
Riemann
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
Following Riemann’s Pioneering Work . . .
. . . von Mangoldt . . . Hadamard . . . de la Vallée Poussin
. . . Hardy . . . Littlewood . . . Selberg . . . Montgomery . . .
The End
Prime Numbers
From Euler to Riemann
A ‘Wonderful Formula’
The End
Many Thanks
for your kind attention
and credits to
Tobias Ebel
for the computer-technical assistance
The End
Appendix with Pictures and Graphics
There are infinitely many prime numbers
Number of Primes up to a Given Size
Number of Primes up to a Given Size
Euler and the ‘Real Zeta Function’
Euler and the ‘Real Zeta Function’
The Complex Number Plane
imaginäre Achse
2i
i
−1
1
2
3
reelle Achse
−i
−i
The Complex Number Plane
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
−i
The Complex Number Plane
imaginäre Achse
z = 2.5 + 2 i
2i
i
−1
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
w = 1 − 0.5 i
−i
The Complex Number Plane
imaginäre Achse
z = 2.5 + 2 i
2i
z+w=
i
−1
3.5 + 1.5 i
1
2
3
reelle Achse
w = 1 − 0.5 i
−i
be visualised as landscapes.
Complex Functions
Georg Friedrich Bernhard Riemann
mplex number plane,
The Riemann Zeta Function
om a pole at s = 1.
The Riemann Zeta Function
The Critical Strip
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
−10 i
−20 i
imaginäre Achse
The Critical Strip
30 i
kritischer Streifen
20 i
10 i
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
−20 i
imaginäre Achse
−20 i
imaginäre Achse
The Critical Strip
30 i
...
kritischer Streifen
20 i
0.5 + i (21.02...)
0.5 + i (14.13...)
10 i
nicht−triviale
Nullstellen
reelle Achse
Pol
−4
−3
−2
−1
1
triviale Nullstellen für s = −2, −4, −6, ...
−10 i
0.5 − i (14.13...)
−20 i
imaginäre Achse
0.5 − i (21.02...)
...
The Critical Strip
The Chebyshev ψ-Function