Download n=1

The Distribution of
Prime Numbers
Date: Monday, March 19th, 2007
Presenter: Anna Yoon
Millennium Prize Problems
“Important classic questions that have resisted to provide solutions over the years”
The first person to solve each problem will be awarded
$100,000 by Clay Mathematics Institute.
1)
P versus NP
2)
The Hodge Conjecture
3)
The Poincare Conjecture
4)
THE RIEMANN HYPOTHESIS
5)
Yang-Mills Existence and Mass Gap
6)
Navier-Stokes Existence and Smoothness
7)
The Birth and Swinnerton-Dyer Conjecture
Ishango Bone
* It is a dark brown bone tool,
the fibula of baboon.
* It was first thought to be a tally stick, as it
has a series of tally marks carved in three
columns.
* However, some scientists have suggested
that the groupings of notches indicated a
mathematical understanding that goes
beyond counting.

It is estimated to originate between 9000 to 6500 BC, and was discovered in
the African area of Ishango (near the Nile River between Uganda and
Congo) in 1960
* It has a series of grouped notches carved in three columns

Central Column :
understanding of
multiplication and
division by 2.
Central Column
Right Column


Right Column:
Odd numbers
Left Column:
Prime Numbers
(11, 13, 17, 19)
Left Column
Ancient Greeks – Pythagoras
(500 ~ 300 BC)

The mathematicians of Pythagoras’s
school were interested in numbers and
their mythical and numerological
properties.

They understood primality, and interested
in perfect and amicable numbers.
( Perfect Number  6
Amicable Numbers  220 and 284 )
Ancient Greeks – Euclid
( 300 BC)

Elements: Book IX, Proposition 20
Suppose you have a finite number of primes.
Call this number m.
Multiply all m primes together and add one.
The resulting number is not divisible by any
of the finite set of primes
Therefore, it must either be prime itself or be divisible by some other
prime that was not included in the finite set.
Either way, there must be at least m+1 primes.
But this argument applies no matter what m is; it applies to m+1 too.
So, there are more primes than any given finite number.
However, ( 2x3x5x7x11x13 ) + 1 = 30031
30 031 = 59 x 509
Sieve of Eratosthenes
( 200 BC)
1) Write out all the numbers from 1 to n :
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
n
2) Take out all the multiples of 2 :
1
2
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
n
3) Take out all the multiples of 3 :
1
2
11
3
5
7
13
17
23
25
31
29
35
41
19
37
43
47
49
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
n
4) Take out all the multiples of 5 :
1
2
11
3
5
7
13
17
19
23
29
31
37
41
43
47
49
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
n
5) Continue the process & eliminate the
non-primes :
1
2
23 29
3
5
7
11
31
37
…. n
13
17
19
After the Greeks, little happened until the 17th century.

Pierre de Fermat
(August 17, 1601 – January 12, 1665)

Fermat Numbers
n
Fn = 22 + 1
n  positive integer
(It doesn’t always produce prime numbers)

Fermat’s Little Theorem
a p = a (mod p)
a p-1 = 1 (mod p)
a  any integer

Marin Mersenne
(September 8, 1588 – September 1, 1648)
Mersenne Prime
Mn = 2n − 1
n > 2, 3, 5, 7, 13, 19, …


Largest Known Prime
as of September 2006 :
M30402457 = 2 30 402 457 - 1
with 9, 808, 358 digits
long !!
Carl Friedrich Gauss
(1777 – 1855)

In 1791, a 14 year old Gauss received a
collection of mathematics books, which
included logarithm table.

In the following year, he observed the
values of the number of primes up to
various powers of ten, and discovered the
pattern and regularity.
N
Number of Primes from 1 to N :
π (N)
On average, how many
primes you need to count
before you reach a prime
number
10
4
2.5
100
25
4.0
1000
168
6.0
10000
1229
8.1
100000
9592
10.4
1000000
78498
12.7

Based on the
observations he
made from his table,
he could estimate
the number of
primes from 1 to N
as roughly as
__N__ .
log N
N
log N

There still seems to be a place for
improvement !!
Gauss :
Integral Logarithmic Function Li(N)
In 1863, Gauss published more accurate
estimate of the number of prime numbers
 Logarithmic Integral Function : Li(N)

Li (N) = ∫2

N
(1 / log u) du
A comparison of the graph of
and Li(N) shows that over a
large range it is barely possible
to distinguish the two.
Integral Logarithmic Function Li(N)
N
Numbers of Primes
from 1 to N :
π (N)
Gauss’s Logarithmic
Integral Function:
N
Li (N) = ∫2 (1 / log u) du
10 3
168
178
10 4
1229
1246
10 5
9592
9630
10 6
78498
78628
10 7
664579
664918
10 8
5761455
5762209
Leonhard Euler
(1707 – 1793)

Euler focused on the nature of prime distribution with ideas in
analysis. In 1737, Euler proved that the sum of the inverses of
the prime numbers is divergent.
∑(1/P)=∞

By doing so, he was able to discover the connection between
Riemann zeta function and prime numbers, known as the
Euler product formula for the Riemann zeta function.
∞
∑ n=1 ( 1 / n s ) = π [ 1 / ( 1 – ( 1 / p s ) ) ]
( Riemann Zeta Function =
Euler Product )

Euler, himself, would not be able to grasp the
full significance of the connection between
Riemann zeta function and prime numbers.

The significance of Euler’s product took
another hundred year to be recognized by
Dirichlet and Riemann.
Bernhard Riemann
(1826 – 1866)

Riemann extended Euler’s zeta function to the entire
complex plane, when studying the distribution of prime
numbers.

In 1859, Riemann published “On the Number of Primes
Less than a Given Magnitude”, the only paper he wrote
on Prime Numbers.

In this paper, Riemann introduced revolutionary ideas
into the subject, that the distribution of prime numbers is
intimately connected with zeros of analytically extended
Riemann zeta function of complex variable.
Riemann Zeta Function:
∞
ζ (s) = ∑ n=1 ( 1 / n s )

1)
2)
Two types of zeros when ζ (s) = 0
Trivial Zeros
Non- Trivial Zeros
_________________________________________________________________________
1)
Trivial Zeros: negative even integers
e.g. S = -2, - 4, - 6 , ……
** Riemann derived the following functional equation : ζ (s) = 2s πs-1 sin(πs/2)Γ(1-s) ζ (1-s)
2) Non -Trivial Zeros: real part = 1/2
Riemann Zeta∞ Function:
ζ (s) = ∑ n=1 ( 1 / n s )
Riemann Hypothesis

Riemann observed that the
set of all non-trivial zeros must
be symmetrical about the
real axis, and the vertical strip
(critical strip) defined
by Re (s) = 1/ 2

At least the first 15 000 000 000
zeros were confirmed using
powerful computer.
Do you want to be a millionaire?
Remember, “Riemann Hypothesis ! ”
Berhnard Riemann said…….
“One would of course like to have a
rigorous proof of this, but I have put
aside the search for such a proof after
some fleeting vain attempts because it is
not necessary for the immediate
objective of my investigation”
Why Prime Numbers?

In the 1970s, Ron Rivest, Adi Shamir and Leonard Adleman,
turned to pursuit of prime numbers into a serious business
application.

They found a way to use the primes to protect our credit card
numbers as well as they travel through the electronic shopping
malls of the global market.

Every time someone places an order on a website, the
computer is using the security provided by the existence of
prime numbers with a hundred digits, this system is called RSA.

Breaking this code depends on the pattern of prime numbers
that we still can not answer.

Business and security agencies are keeping a watchful eye on
the blackboards of the pure mathematicians.