The Distribution of The Distribution of Prime NumbersPrime Numbers

Date: Monday, March 19Date: Monday, March 19thth, 2007, 2007

Presenter: Anna YoonPresenter: Anna Yoon

Millennium Prize ProblemsMillennium Prize Problems““Important classic questions that have resisted to provide solutions over the years”Important classic questions that have resisted to provide solutions over the years”

The first person to solve each problem will be awarded The first person to solve each problem will be awarded $100,000$100,000 by Clay Mathematics Institute. by Clay Mathematics Institute.

1)1) P versus NPP versus NP

2)2) The Hodge ConjectureThe Hodge Conjecture

3)3) The Poincare ConjectureThe Poincare Conjecture

4)4) THE RIEMANN HYPOTHESISTHE RIEMANN HYPOTHESIS

5)5) Yang-Mills Existence and Mass GapYang-Mills Existence and Mass Gap

6)6) Navier-Stokes Existence and SmoothnessNavier-Stokes Existence and Smoothness

7)7) The Birth and Swinnerton-Dyer ConjectureThe Birth and Swinnerton-Dyer Conjecture

Ishango BoneIshango 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 and was discovered in the African area of Ishango (near the Nile River between Uganda and the African area of Ishango (near the Nile River between Uganda and Congo) in 1960Congo) in 1960

* It has a series of grouped notches carved in three columns* It has a series of grouped notches carved in three columns

Central Column : Central Column : understanding of understanding of multiplication and multiplication and division by 2. division by 2.

Right Column: Right Column: Odd numbers Odd numbers

Left Column:Left Column: Prime Numbers Prime Numbers (11, 13, 17, 19)(11, 13, 17, 19)

Central ColumnCentral Column

Right ColumnRight Column

Left ColumnLeft Column

Ancient Greeks – PythagorasAncient Greeks – Pythagoras(500 ~ 300 BC)(500 ~ 300 BC)

The mathematicians of Pythagoras’s The mathematicians of Pythagoras’s school were interested in numbers and school were interested in numbers and their mythical and numerological their mythical and numerological properties.properties.

They understood primality, and interested They understood primality, and interested in perfect and amicable numbers.in perfect and amicable numbers.

( Perfect Number ( Perfect Number 6 Amicable Numbers 6 Amicable Numbers 220 and 284 ) 220 and 284 )

Ancient Greeks – EuclidAncient Greeks – Euclid( 300 BC)( 300 BC)

Elements: Book IX, Proposition 20Elements: Book IX, Proposition 20

Suppose you have a finite number of primes. Suppose you have a finite number of primes.

Call this number m. Call this number m.

Multiply all m primes together and add one. Multiply all m primes together and add one.

The resulting number is not divisible by any The resulting number is not divisible by any

of the finite set of primes of the finite set of primes

Therefore, it must either be prime itself or be divisible by some otherTherefore, it must either be prime itself or be divisible by some other

prime that was not included in the finite set. prime that was not included in the finite set.

Either way, there must be at least m+1 primes. 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. 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.So, there are more primes than any given finite number.

However, ( 2x3x5x7x11x13 ) + 1 = 30031 However, ( 2x3x5x7x11x13 ) + 1 = 30031

30 031 = 59 x 509 30 031 = 59 x 509

Sieve of EratosthenesSieve of Eratosthenes( 200 BC)( 200 BC)

1) Write out all the numbers from 1 to n : 1) Write out all the numbers from 1 to n :

11 22 33 44 55 66 77 88 99 1010

1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030

3131 3232 3333 3434 3535 3636 3737 3838 3939 4040

4141 4242 4343 4444 4545 4646 4747 4848 4949 5050 :: :: :: :: :: :: :: :: :: ::

:: :: :: :: :: :: :: :: :: nn

2) Take out all the multiples of 2 : 2) Take out all the multiples of 2 :

11 22 33 55 77 99

1111 1313 1515 1717 1919

2121 2323 2525 2727 2929

3131 3333 3535 3737 3939

4141 4343 4545 4747 4949 :: :: :: :: :: :: :: :: ::

:::: :: :: :: :: :: :: :: ::

nn

3) Take out all the multiples of 3 : 3) Take out all the multiples of 3 :

11 22 33 55 77

1111 1313 1717 1919

2323 2525 2929

3131 3535 3737

4141 4343 4747 4949 :: :: :: :: :: :: :: :: :: ::

:: :: :: :: :: :: :: :: :: nn

4) Take out all the multiples of 5 : 4) Take out all the multiples of 5 :

11 22 33 55 77

1111 1313 1717 1919

2323 2929

3131 3737

4141 4343 4747 4949 :: :: :: :: :: :: :: :: ::

:::: :: :: :: :: :: :: :: ::

nn

5) Continue the process & eliminate the 5) Continue the process & eliminate the non-primes : non-primes :

11 22 33 55 77 1111 1313 1717 1919

2323 2929 3131 37 37 ….…. nn

After the Greeks, little happened until the 17After the Greeks, little happened until the 17 thth century. century.

Pierre de FermatPierre de Fermat(August 17, 1601 – January 12, 1665)(August 17, 1601 – January 12, 1665)

Fermat NumbersFermat Numbers

FFnn = 2 = 222n n + 1+ 1

n n positive integer positive integer(It doesn’t always produce prime numbers)(It doesn’t always produce prime numbers)

Fermat’s Little TheoremFermat’s Little Theorem

a a p p == a (mod p) a (mod p)

a a p-1p-1 == 1 (mod p) 1 (mod p)

a a any integer any integer

Marin MersenneMarin Mersenne(September 8, 1588 – September 1, 1648) (September 8, 1588 – September 1, 1648)

Mersenne PrimeMersenne Prime

MMnn = 2 = 2nn − 1 − 1

n n > 2, 3, 5, 7, 13, 19, …> 2, 3, 5, 7, 13, 19, …

Largest Known Prime Largest Known Prime as of September 2006 :as of September 2006 :

MM3040245730402457 = 2 = 2 30 402 45730 402 457 - 1 - 1

with 9, 808, 358 digitswith 9, 808, 358 digits

long !!long !!

Carl Friedrich Gauss Carl Friedrich Gauss (1777 – 1855)(1777 – 1855)

In 1791, a 14 year old Gauss received a In 1791, a 14 year old Gauss received a collection of mathematics books, which collection of mathematics books, which included logarithm table. included logarithm table.

In the following year, he observed the In the following year, he observed the values of the number of primes up to values of the number of primes up to various powers of ten, and discovered the various powers of ten, and discovered the pattern and regularity.pattern and regularity.

NN Number of Primes from 1 to N : Number of Primes from 1 to N :

ππ (N) (N)

On average, how many On average, how many primes you need to count primes you need to count before you reach a prime before you reach a prime numbernumber

10 10 44 2.52.5

100100 2525 4.04.0

10001000 168168 6.06.0

1000010000 12291229 8.18.1

100000100000 95929592 10.410.4

10000001000000 7849878498 12.712.7

Based on the Based on the observations he observations he made from his table, made from his table, he could estimate he could estimate the number of the number of primes from 1 to N primes from 1 to N as roughly as as roughly as

____NN____ . .

log N log N

N

N

log

There still seems to be a place for There still seems to be a place for improvement !!improvement !!

Gauss : Gauss : Integral Logarithmic Function Li(N)Integral Logarithmic Function Li(N)

In 1863, Gauss published more accurate In 1863, Gauss published more accurate estimate of the number of prime numbersestimate of the number of prime numbers

Logarithmic Integral Function : Li(N)Logarithmic Integral Function : Li(N)

Li (N) = Li (N) = ∫∫2 2

N N (1 / log u) du (1 / log u) du

A comparison of the graph of A comparison of the graph of and Li(N) shows that over a and Li(N) shows that over a large range it is barely possible large range it is barely possible to distinguish the two.to distinguish the two.

Integral Logarithmic Function Li(N)Integral Logarithmic Function Li(N)NN Numbers of Primes Numbers of Primes

from 1 to N : from 1 to N :

ππ (N) (N)

Gauss’s Logarithmic Gauss’s Logarithmic

Integral Function: Integral Function:

Li (N) = Li (N) = ∫∫2 2

N N (1 / log u) du (1 / log u) du

10 10 33 168168 178178

10 10 44 12291229 12461246

10 10 55 95929592 96309630

10 10 66 7849878498 7862878628

10 10 77 664579664579 664918664918

10 10 88 57614555761455 57622095762209

Leonhard Euler Leonhard Euler (1707 – 1793)(1707 – 1793)

Euler focused on the nature of prime distribution with ideas in Euler focused on the nature of prime distribution with ideas in analysis. In 1737, Euler proved that the sum of the inverses of analysis. In 1737, Euler proved that the sum of the inverses of the prime numbers is divergent. the prime numbers is divergent.

∑ ∑ ( 1 / P ) = ∞( 1 / P ) = ∞

By doing so, he was able to discover the connection between By doing so, he was able to discover the connection between Riemann zeta function and prime numbers, known as the Riemann zeta function and prime numbers, known as the Euler product formula for the Riemann zeta function.Euler product formula for the Riemann zeta function.

∑ ∑ n=1n=1

∞∞ ( 1 / n ( 1 / n ss ) = ) = ππ [ 1 / ( 1 – ( 1 / p [ 1 / ( 1 – ( 1 / p ss ) ) ] ) ) ]

( Riemann Zeta Function = Euler Product ) ( Riemann Zeta Function = Euler Product )

Euler, himself, would not be able to grasp the Euler, himself, would not be able to grasp the full significance of the connection between full significance of the connection between Riemann zeta function and prime numbers.Riemann zeta function and prime numbers.

The significance of Euler’s product took The significance of Euler’s product took another hundred year to be recognized by another hundred year to be recognized by Dirichlet and Riemann.Dirichlet and Riemann.

Bernhard RiemannBernhard Riemann(1826 – 1866)(1826 – 1866)

Riemann extended Euler’s zeta function to the entire Riemann extended Euler’s zeta function to the entire complex plane, when studying the distribution of prime complex plane, when studying the distribution of prime numbers. numbers.

In 1859, Riemann published “In 1859, Riemann published “On the Number of Primes On the Number of Primes Less than a Given Magnitude”Less than a Given Magnitude”, the only paper he wrote , the only paper he wrote on Prime Numbers.on Prime Numbers.

In this paper, Riemann introduced revolutionary ideas In this paper, Riemann introduced revolutionary ideas into the subject, that the distribution of prime numbers is into the subject, that the distribution of prime numbers is intimately connected with zeros of analytically extended intimately connected with zeros of analytically extended Riemann zeta function of complex variable. Riemann zeta function of complex variable.

Riemann Zeta Function:Riemann Zeta Function: ζζ (s) = ∑ (s) = ∑ n=1n=1

∞∞ ( 1 / n ( 1 / n ss ) )

Two types of zeros when Two types of zeros when ζζ (s) = 0 (s) = 0

1)1) Trivial ZerosTrivial Zeros2)2) Non- Trivial ZerosNon- Trivial Zeros__________________________________________________________________________________________________________________________________________________

1)1) Trivial Zeros: negative even integersTrivial Zeros: negative even integers e.g. S = -2, - 4, - 6 , ……e.g. S = -2, - 4, - 6 , …… ** Riemann derived the following functional equation : ** Riemann derived the following functional equation : ζζ (s) = 2s (s) = 2s ππs-1 sin(s-1 sin(ππs/2)s/2)ΓΓ(1-s) (1-s) ζζ (1-s) (1-s)

2) Non -Trivial Zeros: real part = 1/2 2) Non -Trivial Zeros: real part = 1/2

Riemann Zeta Function:Riemann Zeta Function: ζζ (s) = ∑ (s) = ∑ n=1n=1

∞∞ ( 1 / n ( 1 / n ss ) )

Riemann HypothesisRiemann Hypothesis Riemann observed that the Riemann observed that the set of all non-trivial zeros must set of all non-trivial zeros must be symmetrical about the be symmetrical about the real axis, and the vertical strip real axis, and the vertical strip (critical strip) defined (critical strip) defined by Re (s) = 1/ 2 by Re (s) = 1/ 2

At least the first 15 000 000 000 At least the first 15 000 000 000 zeros were confirmed using zeros were confirmed using powerful computer. powerful computer.

Do you want to be a millionaire?Do you want to be a millionaire?

Remember, “Remember, “Riemann Hypothesis Riemann Hypothesis ! ”! ”

Berhnard Riemann said…….Berhnard Riemann said……. “ “One would of course like to have a One would of course like to have a

rigorous proof of this, but I have put rigorous proof of this, but I have put aside the search for such a proof after aside the search for such a proof after some fleeting vain attempts because it is some fleeting vain attempts because it is not necessary for the immediate not necessary for the immediate objective of my investigation”objective of my investigation”

Why Prime Numbers? Why Prime Numbers? In the 1970s, Ron Rivest, Adi Shamir and Leonard Adleman, In the 1970s, Ron Rivest, Adi Shamir and Leonard Adleman,

turned to pursuit of prime numbers into a serious business turned to pursuit of prime numbers into a serious business application.application.

They found a way to use the primes to protect our credit card They found a way to use the primes to protect our credit card numbers as well as they travel through the electronic shopping numbers as well as they travel through the electronic shopping malls of the global market.malls of the global market.

Every time someone places an order on a website, the Every time someone places an order on a website, the computer is using the security provided by the existence of computer is using the security provided by the existence of prime numbers with a hundred digits, this system is called prime numbers with a hundred digits, this system is called RSA.RSA.

Breaking this code depends on the pattern of prime numbers Breaking this code depends on the pattern of prime numbers that we still can not answer.that we still can not answer.

Business and security agencies are keeping a watchful eye on Business and security agencies are keeping a watchful eye on the blackboards of the pure mathematicians. the blackboards of the pure mathematicians.