Transcript The Basel Problem - David Louis Levine

old Swiss banknote honoring Euler
The Basel Problem
Leonhard Euler’s Amazing 1735 Proof that
1 1 1 1

1 2  2  2  2  . . . 
2 3 4 5
6
2
David Levine
Woodinville High School
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
The Riemann Zeta (ζ) Function

The
Greek
letter
zeta
1
 ( n)   n
r 1 r
sum
1 1 1 1 1 1
 1 n  n  n  n  n  n  . . .
2 3 4 5 6 7
• This simple function is very important in the
mathematical fields of analysis and number theory
Bernhard Riemann
(1826-1866)
• 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
The Basel Problem
• In 1650, Pietro Mengoli asked for the value of
1 1 1 1
 (2)  1  2  2  2  2  . . .
2 3 4 5
• 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
1 1 1 1
2
1 2  2  2  2  . . . 
2 3 4 5
6
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
i
• Discovered that e  1  0. 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.
Prime Numbers and Zeta
• 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

1
1 1 1
 ( n)   n  1  n  n  n  . . .
2 3 4
r 1 r
sum
1
 
, n 1
n
product
p prime 1  p
 1  1  1  1  1  1  1 

...
 n 
 n 
 n 
 n 
 n 
 n 
n 
 1  2  1  3  1  5  1  7  1  11  1  13  1  17 
• 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
1 1 1 1
2
1 2  2  2  2  . . . 
?
2 3 4 5
6
• 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
2
6
 1
1 1 1 1
   . . .
22 32 42 52
The Factorial Function
• The factorial function n! is the
product of the numbers 1
through n or
n! 1 2  3  4  5  ...  n 1 n
• For example,
4 ! 1  2  3  4  24
• n! grows very
quickly as n
increases, faster
than most other
functions
• Compare x! to ex
log scale
n
en
n!
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
2
6
 1
1 1 1 1
   . . .
22 32 42 52
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 is
x1 x 2 x 3 x 4 x 5
x
e  1       ...
1! 2! 3! 4! 5!
x1 x 2 x 3 x 4 x 5
 1   

 ...
1 2 6 24 120
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.
Sir Brook Taylor
(1685-1731)
2
6
 1
1 1 1 1
   . . .
22 32 42 52
Taylor Series for sin x
• The sin function’s Taylor series is
x3 x5 x7
sin x   x     ...
3! 5! 7 !
largest degree of each
approximation to sin x
11 7
3
sin x
• 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.
1
13 9 5
2
6
 1
1 1 1 1
   . . .
22 32 42 52
The Fundamental Theorem of Algebra
• Any polynomial of degree n can be written as a
product of exactly n (possibly complex) factors
• Example:
x 4  2 x 3  7 x 2  8 x  12
 x  3x  2x  2x  1
• This degree 4 polynomial has
4 real roots at
x = –2, x = –1, x = 2, and x = 3
roots
2
6
 1
1 1 1 1
   . . .
22 32 42 52
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
sin x  x  ax  bx  c...
• The roots of sin x are at x = 0, ±π , ±2π , ±3π, … so we
write
sin x  Axx   x   x  2 x  2 x  3 x  3  ...
2
2
2
2
2
2
 Axx    x  2  x  3  ...
A is some

real number


this difference of squares has
factors of (x + 3π) and (x – 3π)
and roots at ±3π
2
6
 1
1 1 1 1
   . . .
22 32 42 52
An Exact Expression for sin x
• Our expression for sin x has an unknown factor A




sin x   Ax x 2   2 x 2  2  x 2  3  ...
2
2
1
• Multiply each factor in parentheses by n 2 ,
where n goes up by one each factor
 x 
x 
x 



 ...


sin x   Bx 1  2 1 
1
2 
2 
   2   3  
2
2
2
1
• 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 lim x 0
graph of x and sin x
lim x0
sin x
1
x
0
0
sin x
 1, so
x
1.57
3.14
 02 
sin x 
02 
02 
1 
 ..0. 
1 
the limit
very
closely2 to
without
1...  Bit 1
lim x0
B1as x approaches
B11reaching
2 
2 
x
   2   3  
2
6
 1
1 1 1 1
   . . .
22 32 42 52
Multiply all the Factors
 x 2 
x 2 
x 2 
x 2 
x2 
1 
1 
1 
 ...
sin x   x1  2 1 
2 
2 
2 
2 
   2   3   4   5  
multiply
2
2
2

x3  
x 2  1  x 1  x 1  x  ...
  x  2  1 
 2 2   3 2  4 2  5 2 




multiply
(FOIL)
2
2

x3
x3
x5  


x2  
x
x

  x  2 






 ...
1

1

1

2
2 
2 
2
2 
2 

2   2    3    4   5  

multiply


x3
x3
x3
x5
x5
x5
x7






  x  2 
2
2
2
2
2
2
2
2 
2
2
2

2  3   2   3  2  3   2  3  

each cubic term comes from one x term and one x2 term, with the rest 1’s
multiply remaining factors
x3
x3
x3
x3
x3
x3
 x 2 




 ...  an infinite number of higher degree terms
2
2
2
2
2

2  3  4  5  6 
2
6
 1
1 1 1 1
   . . .
22 32 42 52
Euler’s Genius
• By multiplying all of its factors, we wrote sin x as
x3
x3
x3
x3
sin x   x  2 


 ...  an infinite number of higher degree terms
2
2
2

2  3  4 
• But the Taylor series for sin x is
x3 x5 x7
sin x   x     ...
3! 5! 7 !
• Euler equated the x3 terms from both expressions
multiply by 


2
x3
2
6
• Voila!

1 1 1 1
 2  2  2  ...
2
1 2 3 4
…the result has appeared
as if from nowhere
-Julian Havil
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
2
also 
6
1 1 1
4
• Euler also proved that  4  4  4  4  ... 
1
2 3
90
1
1
1
2 24  76977927   4
and that  26  26  26  26  ... 
1
2
3
27 !
• 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
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-tamsgets-nba-training-from-coach-spo/ downloaded 10/22/10
LeBron James photo from http://www.nikeblog.com/2009/02/05/lebronjames-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