Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Riemann Hypothesis and Prime Numbers
Lee, Chang Min
December 9, 2007
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Outline
1
2
3
4
Brief Description of the Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Theories on Prime Numbers
Prime Counting Function
Prime Number Theorem
Improved PNT
Relation between Riemann Hypothesis and Prime Numbers
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Millenium Prize Problems
Seven problems in mathematics that were stated by the Clay
Mathematics Institute in 2000.
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Millenium Prize Problems
Seven problems in mathematics that were stated by the Clay
Mathematics Institute in 2000.
P verses NP
The Hodge conjecture
The Poincaré conjecture
The Riemann Hypothesis
Yang-Mills existence and mass gap
Navier-Stokes existence and smoothness
The Birch and Swinnerton-Dyer conjecture
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Millenium Prize Problems
Seven problems in mathematics that were stated by the Clay
Mathematics Institute in 2000.
P verses NP
The Hodge conjecture
The Poincaré conjecture
The Riemann Hypothesis
Yang-Mills existence and mass gap
Navier-Stokes existence and smoothness
The Birch and Swinnerton-Dyer conjecture
$ 1,000,000 Prize !
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Statement of the Riemann Hypothesis
Riemann Hypothesis
The real part of any non-trivial zero of the Riemann zeta
function is 12 .
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Statement of the Riemann Hypothesis
Riemann Hypothesis
The real part of any non-trivial zero of the Riemann zeta
function is 12 .
First, what is the zeta function ?
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Formula of the Zeta Function
The formula of the zeta function is,
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Formula of the Zeta Function
The formula of the zeta function is,
ζ(s) = 1 +
=
∞
X
1
1
1
+ s + s + ···
s
2
3
4
n−s
n=1
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Formula of the Zeta Function
The formula of the zeta function is,
ζ(s) = 1 +
=
∞
X
1
1
1
+ s + s + ···
s
2
3
4
n−s
n=1
What are the zeros of the zeta function ?
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Statement
Formula of the Zeta Function
Zeros of the Zeta Function
Zeros of the Zeta Function
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Counting Function
Prime Counting Function π(N)
Number of the prime numbers less than N
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Counting Function
Prime Counting Function π(N)
Number of the prime numbers less than N
Values of π(N) for some large Ns
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Counting Function
Prime Counting Function π(N)
Number of the prime numbers less than N
Values of π(N) for some large Ns
N
1,000
1,000,000
1,000,000,000
1,000,000,000,000
π(N)
168
78,498
50,847,534
37,607,912,018
Table: some values of π(N)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Counting Function
Prime Counting Function π(N)
Number of the prime numbers less than N
Values of π(N) for some large Ns
N
1,000
1,000,000
1,000,000,000
1,000,000,000,000
π(N)
168
78,498
50,847,534
37,607,912,018
Table: some values of π(N)
Isn’t there any rule?
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Ratio of π(N) and N
Let’s think about ratio of π(N) and N
Values of N/ π(N) for some large Ns
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Ratio of π(N) and N
Let’s think about ratio of π(N) and N
Values of N/ π(N) for some large Ns
N
1,000
1,000,000
1,000,000,000
1,000,000,000,000
N/ π(N)
5.9524
12.7392
19.6666
26.5901
Table: some values of N/ π(N)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Ratio of π(N) and N
Let’s think about ratio of π(N) and N
Values of N/ π(N) for some large Ns
N
1,000
1,000,000
1,000,000,000
1,000,000,000,000
N/ π(N)
5.9524
12.7392
19.6666
26.5901
Table: some values of N/ π(N)
It increases by about 7 for each step !
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Number Theorem
When N becomes 1,000 times, N/ π(N) increases by about 7.
So
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Number Theorem
When N becomes 1,000 times, N/ π(N) increases by about 7.
So
N/ π(N) ∼ ln N
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Number Theorem
When N becomes 1,000 times, N/ π(N) increases by about 7.
So
N/ π(N) ∼ ln N
From this, Prime Number Theorem(PNT) is derived.
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Prime Number Theorem
When N becomes 1,000 times, N/ π(N) increases by about 7.
So
N/ π(N) ∼ ln N
From this, Prime Number Theorem(PNT) is derived.
π(N) ∼
Lee, Chang Min
N
ln N
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Improved PNT
Let a function called
logarithmic integral
function
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Improved PNT
Let a function called
logarithmic integral
function
Z x
1
Li(x) =
dt
ln
t
0
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Improved PNT
Let a function called
logarithmic integral
function
Z x
1
Li(x) =
dt
ln
t
0
The improved PNT is
π(N) ∼ Li(x)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Prime Counting Function
Prime Number Theorem
Improved PNT
Improved PNT
Let a function called
logarithmic integral
function
Z x
1
Li(x) =
dt
ln
t
0
The improved PNT is
π(N) ∼ Li(x)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
J function and the Möbius Inversion
Let’s define a function called J
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
J function and the Möbius Inversion
Let’s define a function called J
1 √ 1 √ 1 √
π x + π 3 x + π 4 x + ···
2
3
4
∞
X
1 √i π x
=
i
J(x) = π(x) +
i=1
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
J function and the Möbius Inversion
Let’s define a function called J
1 √ 1 √ 1 √
π x + π 3 x + π 4 x + ···
2
3
4
∞
X
1 √i π x
=
i
J(x) = π(x) +
i=1
Then by the ’Möbius Inversion’,
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
J function and the Möbius Inversion
Let’s define a function called J
1 √ 1 √ 1 √
π x + π 3 x + π 4 x + ···
2
3
4
∞
X
1 √i π x
=
i
J(x) = π(x) +
i=1
Then by the ’Möbius Inversion’,
π(x) = J(x) −
1 √ 1 √ 1 √
J x − J 3 x − J 5 x − ···
2
3
5
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Golden Key
A formula called golden key is
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Golden Key
A formula called golden key is
Z ∞
1
ln ζ(s) =
J(x) x −s−1 dx
s
0
From this, it is possible to express J(x) in terms of ζ(s).
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Golden Key
A formula called golden key is
Z ∞
1
ln ζ(s) =
J(x) x −s−1 dx
s
0
From this, it is possible to express J(x) in terms of ζ(s).
J(x) is expressed by ζ(s) like this
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Golden Key
A formula called golden key is
Z ∞
1
ln ζ(s) =
J(x) x −s−1 dx
s
0
From this, it is possible to express J(x) in terms of ζ(s).
J(x) is expressed by ζ(s) like this
J(x) = Li(x) −
X
ρ
Z
Li(x ) − ln 2 +
ρ
x
∞
t
(t 2
dx
− 1) ln t
(ρ : zeros of the ζ(s))
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Relation between π(x) and ζ(s)
So the logic is,
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Relation between π(x) and ζ(s)
So the logic is,
π(x) can be expressed in terms of J(x)(the Möbius inversion).
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Relation between π(x) and ζ(s)
So the logic is,
π(x) can be expressed in terms of J(x)(the Möbius inversion).
J(x) can be expressed in terms of ζ(s)(from the golden key).
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
J function and the Möbius Inversion
Golden Key
Relation between Prime Numbers and Zeta Function
Relation between π(x) and ζ(s)
So the logic is,
π(x) can be expressed in terms of J(x)(the Möbius inversion).
J(x) can be expressed in terms of ζ(s)(from the golden key).
Therefore, we understand about prime numbers from zeta
function !!
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Improved PNT
Von Koch’s Theorem (1901)
π(x) − Li(x) = O
√
x ln x
And the proof of this theorem begins with
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Improved PNT
Von Koch’s Theorem (1901)
π(x) − Li(x) = O
√
x ln x
And the proof of this theorem begins with
If the Riemann Hypothesis is true,
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Improved PNT
Von Koch’s Theorem (1901)
π(x) − Li(x) = O
√
x ln x
And the proof of this theorem begins with
If the Riemann Hypothesis is true,
If the Riemann Hypothesis is not true, then the world is a very
different place. The whole structure of integers and prime
numbers would be very different to what we could imagine. In
a way, it would be more interesting if it were false, but it
would be a disaster because we’ve built so much round
assuming its truth.
- Peter Sarnak, professor of the Princeton University
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
RSA algorithm
RSA algorithm
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Summary
Relation between Riemann Hypothesis and Prime Numbers
π(x)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Summary
Relation between Riemann Hypothesis and Prime Numbers
π(x) ⇐= J(x)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Summary
Relation between Riemann Hypothesis and Prime Numbers
π(x) ⇐= J(x) ⇐= ζ(s)
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Summary
Relation between Riemann Hypothesis and Prime Numbers
π(x) ⇐= J(x) ⇐= ζ(s)
It is quite hard !
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Summary
Relation between Riemann Hypothesis and Prime Numbers
π(x) ⇐= J(x) ⇐= ζ(s)
It is quite hard !
We are taking Physical Mathematics, so · · ·
Lee, Chang Min
Riemann Hypothesis and Prime Numbers
Brief Description of the Riemann Hypothesis
Theories on Prime Numbers
Relation between Riemann Hypothesis and Prime Numbers
Application of Riemann Hypothesis
Improved PNT
RSA algorithm
Summary
Relation between Riemann Hypothesis and Prime Numbers
π(x) ⇐= J(x) ⇐= ζ(s)
It is quite hard !
We are taking Physical Mathematics, so · · ·
Thank You for Your Attentions !!!
Lee, Chang Min
Riemann Hypothesis and Prime Numbers