Series: Oresme to Euler to $1,000,000 © Joe Conrad Solano Community College December 8, 2012 CMC 3 Monterey Conference

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Presentation transcript:

Series: Oresme to Euler to $1,000,000 © Joe Conrad Solano Community College December 8, 2012 CMC 3 Monterey Conference

Series = … = … =

Series Harmonic Series: Nicole Oresme (ca – 1382)

Pietro Mengoli (1626 – 1686)

Jacob Bernoulli (1654 – 1705) p-Series:

Basel Problem “If anyone finds and communicates to us that which thus far has eluded our efforts, great will be our gratitude.” - Jacob Bernoulli, 1689

Enter Euler! Euler ( ) in 1735 computed the sum to 20 decimal places. “Quite unexpectedly I have found an elegant formula involving the quadrature of the circle.”

Euler’s First “Proof” Recall that if P(x) is a n th degree polynomial with roots a 1, a 2, …, a n, then P(x) can be factored as for some constant A.

Euler let P(x) be Note: xP(x) = sin(x), so So if a is a root of P(x), then sin(a) = 0 which implies that a = ± , ±2 , ±3 , …

So, we can factor P(x) as Letting x = 0, we get B = 1.

Extending this argument, Euler got: In 1750, he generalized this to …

But, first!

Bernoulli discovered how to compute these in general:

“…it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum ”

What about ? The first 20 Bernoulli numbers:

What did Euler know and when? He knew Bernoulli’s work. He knew his p-series sums (1735). He knew the Euler-MacLaurin formula (1732):

He knew the Taylor series for many functions. Somehow, he noticed that the Bernoulli numbers tied these things together.

Appear in Taylor series:

Euler-Maclaurin became:

What about ? Nobody knows the exact sum! Roger Apéry (1916 – 1994) proved this is irrational in 1977.

Where to next? Being calculus, we define a function: This function is defined for all real x > 1.

Bernhard Riemann (1826 – 1866) Define a function: where s complex.

This function can be extended to all the complex numbers except s = 1. Riemann’s Functional Equation: Note:, n a natural number

Question: Are there any other zeros? Riemann found three: ½ i ½ i ½ i

The Riemann Hypothesis All the nontrivial zeros of the zeta function have real part equal to ½.

Carl Siegel (1896 – 1981)

What is known? All nontrivial zeros have 0 < Rez < 1. If z is a zero, then so is its conjugate. There are infinitely many zeros on the critical line. At least 100 billion zeros have been found on the critical line. The first 2 million have been calculated. This verifies the RH up to a height of about 29.5 billion.

What is known? The 100,000 th is ½ + 74, i. The 10,000,000,000,000,000,010,000 th is ½+1,370,919,909,931,995,309, i Andrew Odlyzko

In 2000, the Clay Institute of Mathematics offered a prize for solving the Riemann Hypothesis: $1,000,000

Main Sources Julian Havil, Gamma, Princeton University Press, Princeton, NJ, William Dunham, Euler: The Master of Us All, MAA, Ed Sandifer, How Euler Did It: Bernoulli Numbers, MAA Online, Sept