The Basel problem is a famous problem in number theory, first posed by Pietro Mengoli in 1644, and solved by Leonhard Euler in 1735. The problem had withstood the attacks of the leading mathematicians of the day, so Euler's solution gained him immediate notoriety at the age of 28. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family, who unsuccessfully attacked the problem. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... // Events February to August - Explorer Abel Tasmans second expedition for the Dutch East India Company maps the north coast of Australia. ... Leonhard Euler by Emanuel Handmann Leonhard Euler [oilÉ™r] (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. ... Events April 16 - The London premiere of Alcina by George Frideric Handel, his first the first Italian opera for the Royal Opera House at Covent Garden. ... Bernhard Riemann. ... 1859 is a common year starting on Saturday. ... On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. ... In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... Location within Switzerland Basel (English traditionally: Basle , German: Basel , French Bâle , Italian Basilea ) is Switzerlands third most populous city (188,000 inhabitants in the canton of Basel-City as of 2004; the 690,000 inhabitants in the conurbation stretching across the immediate cantonal and national boundaries made Basel... Eight (8) members of the Bernoulli family were mathematicians: Daniel Bernoulli (1700–1782) Nicolaus II Bernoulli (1695–1726), his elder brother Johann Bernoulli (1667–1748), father of Daniel and Nicolaus II Jakob Bernoulli (also James or Jacques) (1654–1705), brother of Johann Nicolaus I Bernoulli (1687–1759), nephew of Jakob...

The Basel problem asks for the precise sum of the reciprocals of the squares of the positive integers, i.e. the precise sum of the infinite series Addition is one of the basic operations of arithmetic. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, a series is a sum of a sequence of terms. ...

The series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series, (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be π²/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof. In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... Jump to: navigation, search Lower-case pi The mathematical constant Ï€ is the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. ... Events April 16 - The London premiere of Alcina by George Frideric Handel, his first the first Italian opera for the Royal Opera House at Covent Garden. ...

Euler attacks the problem

Euler's original "derivation" of the value π²/6 is clever and original. He essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification, but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

To follow Euler's argument, recall the Taylor series expansion of the sine function Jump to: navigation, search As the degree of the Taylor series rises, it approaches the correct function. ... Jump to: navigation, search In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Dividing through by x, we have

Now, the roots (zeros) of sin(x)/x occur precisely at x = ±nπ, where n = 1, 2, 3, ... Let us assume we can express this infinite series as a product of linear factors given by its roots, just as we do for finite polynomials:

If we formally multiply out this product and collect all the x² terms, we see that the x² coefficient of sin(x)/x is

But from the original infinite series expansion of sin(x)/x, the coefficient of x² is −1/(3!) = −1/6. These two coefficients must be equal; thus,

Multiplying through both sides of this equation by −π² gives the sum of the reciprocals of the positive square integers.

The Riemann zeta function

The Riemann zeta function ζ(s) is one of the most important functions in mathematics, because of its relationship to the distribution of the prime numbers. The function is defined for any complex number s with real part > 1 by the following formula: In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Jump to: navigation, search In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

Taking s = 2, we see that ζ(2) is equal to the sum of the reciprocals of the squares of the positive integers:

How do we know it converges at all? We can demonstrate this with the following inequality:

This gives us the upper bound ζ(2) < 2, but the exact value ζ(2) = π²/6 was unknown for some time, until Leonhard Euler computed it in 1735. It can be shown that ζ(s) has a nice expression in terms of the Bernoulli numbers whenever s is a positive even integer. Jump to: navigation, search Lower-case pi The mathematical constant Ï€ is the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. ... Leonhard Euler by Emanuel Handmann Leonhard Euler [oilÉ™r] (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. ... Events April 16 - The London premiere of Alcina by George Frideric Handel, his first the first Italian opera for the Royal Opera House at Covent Garden. ... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

A rigorous proof

The following argument proves the identity ζ(2) = π²/6, where ζ(s) is the Riemann zeta function. It is by far the simplest proof yet available; while most proofs utilise results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (although a single limit is taken at the end). In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ... In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a function of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... Jump to: navigation, search This article may be too technical for most readers to understand. ... Multivariate calculus is a means of analyzing deterministic systems with multiple degrees of freedom. ... For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... Jump to: navigation, search In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

History of this proof

The origin of the proof is unclear. It appeared in the journal Eureka in 1982, attributed to John Scholes, but Scholes claims he learned the proof from Peter Swinnerton-Dyer, and in any case he maintains the proof was "common knowledge at Cambridge in the late 1960s". Jump to: navigation, search 1982 is a common year starting on Friday of the Gregorian calendar. ... Sir Henry Peter Francis Swinnerton-Dyer (known as Peter Swinnerton-Dyer) is a English mathematician specialising in number theory at Cambridge University. ... The University of Cambridge is the second-oldest university in the English-speaking world, with one of the most selective sets of entry requirements in the United Kingdom. ...

What you need to know

To understand the proof, you will need to understand the following facts:

• De Moivre's formula states that for any real number x and any integer n,

(cosx + isinx)ⁿ = cos(nx) + isin(nx).

Proof: This can be proved from Euler's formula; see the article for more details.

• The binomial theorem, which states that for any real numbers x and y and any nonnegative integer n,

where we have the binomial coefficients

(See factorial)

Proof: This requires mathematical induction and some properties of the binomial coefficients.

• The function cot² x is one-to-one on the interval (0, π/2).
  • Proof: Suppose cot² x = cot² y for some x, y in the interval (0, π/2). Using the definition of cotangent cot x = (cos x)/(sin x) and the trigonometric identity cos² x = 1 − sin² x, we see that (sin² x)(1 − sin² y) = (sin² y)(1 − sin² x). Subtracting (sin² x)(sin² y) from each side, we have sin² x = sin² y. Since the sine function is always nonnegative on the interval (0, π/2), this means sin x = sin y, but it is geometrically evident (by looking at the unit circle, e.g.) that the sine function is 1-1 on the interval (0, π/2), so that x = y.

• If p(t) is a polynomial of degree m, then p has no more than m distinct roots.
  • Proof: This is a consequence of the fundamental theorem of algebra.

• If p(t) = a_mt^m + a_{m − 1}t^{m − 1} + ... + a₁t + a₀, where a_m ≠ 0, then the sum of the roots of p (counting multiplicities) is −a_{m − 1}/a_m.
  • Proof: If a_m = 1, then p(t) = product of all (t − s), where s ranges over all roots of p. Expanding this product, we see the coefficient of t^{m − 1} is minus the sum of all the roots. If a_m ≠ 1, then we can divide each term by it, obtaining a new polynomial with the same roots, whose leading coefficient is now 1; applying the preceding argument shows that the sum of the roots of the original p(t) = sum of the roots of this new polynomial = −a_{m − 1}/a_m.

• The trigonometric identity csc² x = 1 + cot² x.
  • Proof: This follows from the fundamental identity 1 = sin² x + cos² x after dividing through by sin² x.

• For any real number x with 0 < x < π/2, we have the inequalities cot² x < 1/x² < csc² x.
  • Proof: First note that 0 < sin x < x < tan x. This can be seen by considering the following picture:

De Moivres formula states that for any real number x and any integer n, The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Jump to: navigation, search The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Jump to: navigation, search In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n roots (zeroes), counted with multiplicity. ... Jump to: navigation, search In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Define several trig functions from unit circle. ...

Now, take the reciprocal of everything and square. Remember that the inequality switches direction.

• Let a, b, and c be any real numbers, with a and c not both zero; then the limit of the function (am + b)/(am + c) as m approaches infinity is 1.
  • Proof: Divide each term by m, and get (a + b/m)/(a + c/m). If we divide a fixed number by a larger and larger number, the quotient approaches zero; thus, both the numerator and the denominator above tend to a, and so their quotient tends to 1.

• The squeeze theorem, which states that if a function is "squeezed" between two other functions, and each of those two functions approach a common limit, then the "squeezed" function also approaches that same limit.
  • Proof: See the article for a thorough discussion and proof.

In calculus, the squeeze theorem, (also known as the pinching theorem or sandwich theorem) is a theorem regarding the limit of a function. ...

The proof

The main idea behind the proof is to bound the partial sums

between two expressions, each of which will tend to π²/6 as m approaches infinity. The two expressions are derived from identities involving the cotangent and cosecant functions. These identities are in turn derived from De Moivre's formula, and we now turn to establishing these identities.

Let x be a real number with 0 < x < π/2, and let n be a positive integer. Then from De Moivre's formula and the definition of the cotangent function, we have

From the binomial theorem, we have

Combining the two equations and equating imaginary parts gives the identity

We take this identity and set n = 2m + 1, where m is a positive integer, and x = rπ/(2m + 1), where r = 1, 2, ..., m. Then nx = rπ, so that sin(nx) = 0, and so

This equation holds for each of the values x = rπ/(2m + 1), where r = 1, 2, ..., m. These values of x are distinct numbers strictly between 0 and π/2. Since the function cot²(x) is 1-1 on the interval (0, π/2), the numbers cot²(x) = cot²(rπ/(2m + 1)) are therefore distinct for r = 1, 2, ..., m. But by the above equation, each of these m distinct numbers is a root of the mth degree polynomial

This means that the numbers x = cot²(rπ/(2m + 1)), for r = 1, 2, ..., m are precisely the roots of the polynomial p(t). But we can calculate the sum of the roots directly by examining the coefficients, and the comparison shows that

Substituting the identity csc² x = cot² x + 1, we have

Now consider the inequality cot² x < 1/x² < csc² x. If we add up all these inequalities for each of the numbers x = rπ/(2m + 1), and if we use the two identities above, we get

Multiplying through by (π/(2m + 1))², this becomes

As m approaches infinity, the left and right hand expressions each approach π²/6, so by the squeeze theorem,

and this completes the proof. Q.E.D. Jump to: navigation, search Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. ...

References

• Number Theory: An Approach Through History, Andre Weil, Springer, ISBN 0817631410
• Euler: The Master of Us All, William Dunham, MAA, ISBN 0883853280
• Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, John Derbyshire, Joseph Henry Press, ISBN 0309085497
• Proofs From the Book, Martin Aigner, Gunter Ziegler, Springer, ISBN 3540678654
• Riemann's Zeta Function, Harold M. Edwards, Dover, ISBN 0486417409

External links

• A page dedicated to ζ(2)
• Euler's solution of the Basel problem -- the longer story (PDF)
• How Euler did it (PDF)
• The infinite series of Euler and the Bernoulli's spice up a calculus class (PDF)
• Fourteen proofs of the evaluation of ζ(2), compiled by Robin Chapman
  • PDF version
  • PS version
  • DVI version