Short Powerpoint presentation on this book.

1. Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics By John Derbyshire, Joseph Henry Press, 2003

2. The Riemann Hypothesis - 1859 All non-trivial zeros of the zeta function have real part one-half Hilbert’s 8th Great Problem 1900 Congress of Mathematicians.

3. Several Looks at the Riemann Zeta Function

4. Two Looks at the Riemann Zeros

5. The Golden Key - Euler 1737 Sieve of Eratosthenes 230 BCE Primes and all their multiples are eliminated from the RHS

6. What happens if we do this for every Prime to  ?

7. The Prime Number Theorem  (n) ~ N/ln(N) The probability that N is prime is: 1/ln(N) The Nth Prime Number is N  ln(N)  (n) is the prime number counting function. Gauss -1849

8. Improved Prime Number Theorem  (n) ~ Li(N) Li(n) ~ N  ln(N) This is a precise, unproven expression for  (n), given in Riemann’s 1859 paper. Proved in 1896, independently by Jacques Hadamard and Charles de la Vallee Poussin. Based on all non-trivial zeros having real part less than 1.

9. Of note, Hardy, 1914, infinitely many of the Riemann non-trivial zeros have real part 1/2. Littlewood, 1914, Li crosses  (x) infinitely many times. Bays and Hudson showed the lowest known violations in 2000 around 1.39822 * 10 316

10. What does this have to do with Riemann zeta zeros? The prime number counting function,  (x), can be expressed in terms of  (s).  (x) belongs to number theory,  (s) to analysis and calculus. Analytic Number Theory is born. Von Koch, 1901, If the Hypothesis is true then:  (x) = Li(x) + O(x 1/2 ln(x))

11. The Mobius Function -  (n) For the natural numbers:  (1) = 1  (n)=0, if n has a square factor.  (n)=-1, if n is prime or is the product of an odd number of primes.  (n)=1, if n is the product of an even number of different primes. Merten’s Function:

12. Prove: And the Riemann Hypothesis is true! Another weird identity.

13. Write down the Natural Numbers. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12… Eliminate numbers that have a square prime factor. 2, 3, 5, 6, 7, 10, 11 Label a number tails if there is an odd number of primes and heads if there are an even number. T, T, T, H, T, H, T If we could prove a 50-50 Distribution with deviation, we would have proved the Riemann Hypothesis.

14. What about Physics? Statistical Quantum Mechanics and Quantum Chaos use GUE, the Gaussian Unitary Ensemble. This is the collectionof Hermitian Matrices whose members contain a Gaussian Normal Distribution of Random Numbers. The pair-wise correlation of the eigenvalues of such a matrix goes as 1-(sin(  u)/  u) 2 and so does the distribution of Riemann’s Zeta Zeroes! Berry, 1986, argues that a quantum system with such an operator would model a dynamical quantum chaotic system Freeman Dyson/Hugh Montgomery - Princeton 1972 Berry -1986