The Basel Problem

Leonhard Euler’s Amazing 1735 Proof that

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David LevineWoodinville High School

old Swiss banknote honoring Euler

Beautiful Mathematics

• Would you want to play basketball if all you ever saw of it was drills, and never the fun of an actual game?

• Today you’ll get to watch one of history’s greatest mathematical artists, Euler (“oiler”), at play

• We’ll start with one of math’s snazziest bits of finesse – the Riemann zeta function

• This simple function is very important in the mathematical fields of analysis and number theory

The Riemann Zeta (ζ) Function

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Bernhard Riemann(1826-1866)

The Greek letter zeta

• One of the most important unsolved problems in mathematics is the Riemann Hypothesis, which states that all the complex roots of the zeta function have a real component equal to ½

• Solving the Riemann Hypothesis would lead to a fundamentally greater understanding of how prime numbers are distributed among the integers

sum

The Basel Problem

• In 1650, Pietro Mengoli asked for the value of

• This was the famous Basel Problem

• By 1665, ζ(2) was known to be about 1.645• In 1735, the Swiss mathematician Leonhard Euler

calculated ζ(2) to 20 decimal places (without a calculator!) and proved, as we will also, that

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ζ(2) ≈ 1.64493406684822643647

• Discovered that . This identity combines the five most basic constants in math in the simplest possible way!

• Euler introduced the concept of a function and function notation.

Leonhard Euler(1708-1783)

• Did important work in: number theory, artillery, northern lights, sound, the tides, navigation, ship-building, astronomy, hydrodynamics, magnetism, light, telescope design, canal construction, and lotteries

• One of the most important mathematicians of all time

• It’s said that he had such concentration that he would write his research papers with a child on each knee while the rest of his thirteen children raised uninhibited pandemonium all around him

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Prime Numbers and Zeta

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• Euler also proved a profound formula that equates a sum of powers of all the natural numbers with a product of powers of all the prime numbers

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• This formula’s proof isn’t hard to understand, but let’s turn our focus to the main atttraction!

Euler’s Really Cool Proof

• How did Euler prove that ?

• The next eight slides wind through several areas of mathematics to reach Euler’s amazing conclusion

• Watch Euler’s brilliance and the proof’s beauty• Euler’s proof begins with an infinite polynomial called

a Taylor series, which you’ll see in calculus

• First, you need to know what the factorial function is

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The Factorial Function• The factorial function n! is the

product of the numbers 1 through n or

• For example,

nnn 1...54321!

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n n! en

1 1 2.7

2 2 7.4

3 6 20.1

4 24 54.6

5 120 148.4

6 720 403.4

7 5040 1096.6

8 40320 2981.0

9 362880 8103.1

10 3628800 22026.5

11 39916800 59874.1

12 479001600 162754.8

13 6227020800 442413.4

• n! grows very quickly as n increases, faster than most other functions

• Compare x! to ex

log scale

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Taylor Series

• In 1715, Brook Taylor found a general way to write any smooth function as an infinite degree polynomial

• For example, the Taylor series for ex isSir Brook Taylor

(1685-1731)

The exponential function is in blue, and the sum of the first n + 1 terms of its Taylor series at 0 is in red. As n increases, the Taylor series gets more accurate.

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Taylor Series for sin x

• The sin function’s Taylor series is

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sin753

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sin x

largest degree of eachapproximation to sin x

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• As the degree of the Taylor polynomial rises, its graph approaches sin x. This image shows sin x (in black) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

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The Fundamental Theorem of Algebra

• Any polynomial of degree n can be written as a product of exactly n (possibly complex) factors

• Example:

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• This degree 4 polynomial has 4 real roots atx = –2, x = –1, x = 2, and x = 3

roots

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• The roots of sin x are at x = 0, ±π , ±2π , ±3π, … so we write

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The Roots of sin x• The Taylor series for sin x is a

polynomial• The Fundamental Theorem of

Algebra says that therefore sin x can be written as a product of its roots

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A is some real number

this difference of squares has factors of (x + 3π) and (x – 3π) and roots at ±3π

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• The factors still have the same roots (zeros), but now B is a different real number. What is B?

• In first year calculus we prove that

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sinlim 2

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• Our expression for sin x has an unknown factor A

• Multiply each factor in parentheses by , where n goes up by one each factor

An Exact Expression for sin x

1sin

lim 0 x

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graph of x and sin x

the limit as x approaches very closely to 0 without reaching it

1sin

lim 0 x

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, so

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Multiply all the Factors

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multiply(FOIL)

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each cubic term comes from one x term and one x2 term, with the rest 1’s

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multiply remaining factors

• Euler equated the x3 terms from both expressions

Euler’s Genius

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sin 2

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sin753

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• By multiplying all of its factors, we wrote sin x as

• But the Taylor series for sin x is

…the result has appearedas if from nowhere

-Julian Havil

multiply by 3

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• Voila!

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Too Good to be True?

• Did you think that some parts of this proof were fuzzy?

• Euler lived before mathematicians could rigorously complete this proof using modern techniques of real analysis

• Does the Fundamental Theorem of Algebra really work for infinite degree polynomials?

• Is it really OK to equate the infinite series of cubic terms?

• Euler wasn’t wrong, but his proof wasn’t complete

Interesting Tidbits

• The probability that any two random positive integers have no common factors (are coprime) is also

• Euler also proved that

and that

• Euler found a general formula for ζ(n) for every even value of n

• Three hundred years later, nobody has found a formula for ζ(n) for any odd value of n

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Slide Notes• This presentation was inspired by and based in large part on the book

Gamma by Julian Havil, Princeton University Press, 2003• Unless listed below, the photographs are in the public domain because their

copyrights have expired or because they are in the Wikipedia commons.• The two Taylor series graphs are in the Wikipedia commons. I annotated

the sine graph. I made the other graphs.• Basketball drill photo from http://ph.yfittopostblog.com/2010/08/10/feu-tams-

gets-nba-training-from-coach-spo/ downloaded 10/22/10• LeBron James photo from http://www.nikeblog.com/2009/02/05/lebron-

james-drops-52-points-triple-double-respect-of-knicks/ downloaded 10/22/10

• Waterfall image from http://grandcanyon.free.fr/images/cascade/original/Proxy Falls, Cascade Range, Oregon.jpg downloaded 10/24/10 and was reflected horizontally and lightened