Leonhard Euler’s Amazing 1735 Proof that The Basel Problem old Swiss banknote honoring Euler Leonhard Euler’s Amazing 1735 Proof that 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 sum 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 Solving the Riemann Hypothesis would probably lead to a fundamentally greater understanding of how prime numbers are distributed

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 ζ(2) ≈ 1.64493406684822643647

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 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.

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 sum product The proof of the formula on this slide is less technical but probably more complex than the proof that zeta(2) = pi^2/6 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 You’ll need to see the proof several times and reflect on it to understand all the steps

The Factorial Function The factorial function n! is the product of the numbers 1 through n or For example, n n! en 1 2.7 2 7.4 3 6 20.1 4 24 54.6 5 120 148.4 720 403.4 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 e^x is graphed as a straight line because it’s an exponential function and the y axis is a logarithmic scale log scale

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 Sir Brook Taylor (1685-1731) Each term brings the approximation just a little bit closer to e^x because the terms decrease as the degree increases. 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.

Taylor Series for sin x The sin function’s Taylor series is largest degree of each approximation to sin x The sin function’s Taylor series is 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

The Fundamental Theorem of Algebra Any polynomial of degree n can be written as a product of exactly n (possibly complex) factors Example: This degree 4 polynomial has 4 real roots at x = –2, x = –1, x = 2, and x = 3 roots

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 The roots of sin x are at x = 0, ±π , ±2π , ±3π, … so we write Question: Why is there a factor of x after the A? Answer: it corresponds to the root at x = 0 A is some real number this difference of squares has factors of (x + 3π) and (x – 3π) and roots at ±3π

An Exact Expression for sin x Our expression for sin x has an unknown factor A Multiply each factor in parentheses by , where n goes up by one each factor graph of x and sin x The factors still have the same roots (zeros), but now B is a different real number. What is B? Where did the x go in the second equation? It was divided by x in the limit of sin(x)/x. Because the limit of sin x as x approaches 0 is 1, for very small angles of x, sinx = x. This “trick” pops up unexpectedly in more advanced math problems – don’t forget it! In first year calculus we prove that , so the limit as x approaches very closely to 0 without reaching it

Multiply all the Factors (FOIL) multiply In the final expression, there are no factors, only terms that are added and subtracted. There is one degree 3 term for each natural number 1, 2, 3,, 4, … There is one degree 5 term for every pair of natural numbers (see the degree five terms in the slide). There is one degree seven term for every triplet of natural numbers, etc. Terms with degree 1, 5, 9, 13, … are added and terms with degrees 3, 7, 11, 15, … are added. multiply remaining factors each cubic term comes from one x term and one x2 term, with the rest 1’s

…the result has appeared Euler’s Genius By multiplying all of its factors, we wrote sin x as But the Taylor series for sin x is Euler equated the x3 terms from both expressions Summing up: we wrote the sin function as the product of its roots and found the cubic terms of the product. We equated those cubic terms with the cubic terms in the Taylor series for sin x. Multiply by –pi^2/x^3 and we have our proof! multiply by …the result has appeared as if from nowhere -Julian Havil Voila!

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

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