The Distribution of Prime Numbers Date Monday March

The Distribution of Prime Numbers Date: Monday, March 19 th, 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. l 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 l Central Column : understanding of multiplication and division by 2. Central Column Right Column l l Right Column: Odd numbers Left Column: Prime Numbers (11, 13, 17, 19) Left Column

Ancient Greeks – Pythagoras (500 ~ 300 BC) l The mathematicians of Pythagoras’s school were interested in numbers and their mythical and numerological properties. l They understood primality, and interested in perfect and amicable numbers. ( Perfect Number 6 Amicable Numbers 220 and 284 )

Ancient Greeks – Euclid ( 300 BC) l 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, ( 2 x 3 x 5 x 7 x 11 x 13 ) + 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 primes : 1 2 23 29 3 5 7 11 31 37 …. n 13 17 non- 19

After the Greeks, little happened until the 17 th century. l Pierre de Fermat (August 17, 1601 – January 12, 1665) Ø Fermat Numbers n F n = 22 + 1 n positive integer (It doesn’t always produce prime numbers) Ø Fermat’s Little Theorem a p = a (mod p) a p-1 = 1 (mod p) a any integer l Marin Mersenne (September 8, 1588 – September 1, 1648) Mersenne Prime Mn = 2 n − 1 n > 2, 3, 5, 7, 13, 19, … Ø Ø Largest Known Prime as of September 2006 : M 30402457 = 2 30 402 457 - 1 with 9, 808, 358 digits long !!

Carl Friedrich Gauss (1777 – 1855) l In 1791, a 14 year old Gauss received a collection of mathematics books, which included logarithm table. l 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

l 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

l 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 l Logarithmic Integral Function : Li(N) l Li (N) = ∫ 2 l 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) l 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)=∞ l 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 )

l Euler, himself, would not be able to grasp the full significance of the connection between Riemann zeta function and prime numbers. l The significance of Euler’s product took another hundred year to be recognized by Dirichlet and Riemann.

Bernhard Riemann (1826 – 1866) l Riemann extended Euler’s zeta function to the entire complex plane, when studying the distribution of prime numbers. l In 1859, Riemann published “On the Number of Primes Less than a Given Magnitude”, the only paper he wrote on Prime Numbers. l 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) = ∑ l 1) 2) ∞ n=1 ( 1/ns) 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) = 2 s 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 l 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 l At least the first 15 000 000 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? l In the 1970 s, Ron Rivest, Adi Shamir and Leonard Adleman, turned to pursuit of prime numbers into a serious business application. l 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. l 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. l Breaking this code depends on the pattern of prime numbers that we still can not answer. l Business and security agencies are keeping a watchful eye on the blackboards of the pure mathematicians.