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Oresme. to Euler to $1,000,000 . © . Joe . Conrad. Solano Community College. December 8, 2012. CMC. 3. Monterey Conference. joseph.conrad@solano.edu. Series. = 0.3 + 0.03 + 0.003 + 0.0003 + …. . ID: 211272

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Series: Oresme to Euler to $1,000,000

©

Joe

Conrad

Solano Community College

December 8, 2012

CMC

3

Monterey Conference

joseph.conrad@solano.edu

Slide2Series

= 0.3 + 0.03 + 0.003 + 0.0003 + … = 0.3333… =

Slide3Series

Harmonic Series: Nicole Oresme (ca. 1323 – 1382)

Slide4Pietro

Mengoli (1626 – 1686)

Slide5Jacob Bernoulli (1654 – 1705)

p-Series:

Slide6Basel Problem

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

Slide7Enter Euler!

Euler (1707 - 1783) in 1735 computed the sum to 20 decimal places.“Quite unexpectedly I have found an elegant formula involving the quadrature of the circle.”

Slide8Euler’s First “Proof”

Recall that if P(x) is a nth degree polynomial with roots a1, a2, …, an, then P(x) can be factored as for some constant A.

Slide9Euler let P(x) beNote: xP(x) = sin(x), soSo if a is a root of P(x), then sin(a) = 0which implies that a = ±, ±2, ±3, …

Slide10So, we can factor P(x) asLetting x = 0, we get B = 1.

Slide11Slide12

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

Slide13But, first!

Slide14Bernoulli discovered how to compute these in general:

Slide15Slide16

“…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

91 409 924 241 424 243 424 241 924 242 500.”

Slide17What about ?The first 20 Bernoulli numbers:

Slide18What did Euler know and when?

He knew Bernoulli’s work.He knew his p-series sums (1735).He knew the Euler-MacLaurin formula (1732):

Slide19He knew the Taylor series for many functions.

Somehow, he noticed that the Bernoulli numbers tied these things together.

Slide20Appear in Taylor series:

Slide21Euler-Maclaurin became:

Slide22Slide23

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

Slide24Where to next?

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

Slide25Bernhard Riemann (1826 – 1866)Define a function:where s complex.

Slide26This function can be extended to all the complex numbers except s = 1.Riemann’s Functional Equation:Note: , n a natural number

Slide27Slide28

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Question: Are there any other zeros?

Riemann found three:

½

+

14.1347

i

½ +

21.0220

i

½

+

25.0109

i

Slide32The Riemann Hypothesis

All the nontrivial zeros of the zeta

function have real part equal to ½.

Slide33Carl Siegel

(1896 – 1981)

Slide34What is known?

All nontrivial zeros have 0 <

Re

z

< 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.

Slide35What is known?

The 100,000th is ½ + 74,920.8275i. The 10,000,000,000,000,000,010,000th is ½+1,370,919,909,931,995,309,568.3354i Andrew Odlyzko

Slide36In 2000, the Clay Institute of Mathematics offered a prize for solving the Riemann Hypothesis:

$1,000,000

Slide37Main Sources

Julian

Havil

,

Gamma

, Princeton University Press

,

Princeton, NJ,

2003.

William Dunham,

Euler: The Master of Us All

, MAA, 1999.

Ed

Sandifer

, How Euler Did It: Bernoulli Numbers,

MAA Online,

Sept. 2005

.

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