The
Basel problem is a famous problem in
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, first posed by
Pietro MengoliPietro Mengoli was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647...
in 1644, and solved by
Leonhard EulerLeonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is in English ; the common English pronunciation is incorrect....
in 1735. Since the problem had withstood the attacks of the leading
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
s of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by
Bernhard Riemannwas an influential German mathematician who made contributions to analysis and differential geometry, some of them enabling the later development of general relativity.-Early life:...
in his seminal 1859 paper
On the Number of Primes Less Than a Given Magnitude die Anzahl der Primzahlen unter einer gegebenen is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akadademie der Wissenschaften zu Berlin...
, in which he defined his
zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers...
and proved its basic properties. The problem is named after
BaselBasel is Switzerland's third most populous city . With 830000 inhabitants in the tri-national metropolitan area , Basel is Switzerland's second-largest urban area....
, hometown of Euler as well as of the Bernoulli family, who unsuccessfully attacked the problem.
The Basel problem asks for the precise
summationSummation is the addition of a set of numbers; the result is their sum or total. An interim or present total of a summation process is termed the running total. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. An infinite sum is a...
of the
reciprocalsIn mathematics, a multiplicative inverse or reciprocal for a number x, denoted by
1⁄
x or x
−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of x is also called the reciprocal of x...
of the
squaresIn mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square...
of the
natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s, i.e. the precise sum of the
infinite seriesIn mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the...
:
The series is approximately equal to
1.644934 . The Basel problem asks for the
exact sum of this series (in
closed formIn mathematics, an expression is said to be a closed-form expression if, and only if, it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
), as well as a
proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
that this sum is correct. Euler found the exact sum to be 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.
Euler attacks the problem
Euler's original "derivation" of the value is clever and original. He essentially extended observations about finite
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
s 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.
To follow Euler's argument, recall the
Taylor seriesIn mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after the English mathematician Brook Taylor...
expansion of the
sine functionIn mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
Dividing through by
x, we have
Now, the roots (zeros) of sin(
x)/
x occur precisely at where
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 gives the sum of the reciprocals of the positive square integers.
The Riemann zeta function
The
Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers...
is one of the most important functions in mathematics, because of its relationship to the distribution of the
prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
s. The function is defined for any
complex numberA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
s with real part > 1 by the following formula:
Taking
s = 2, we see that is equal to the sum of the reciprocals of the squares of the positive integers:
Convergence can be proven with the following inequality:
This gives us the upper bound , and because the infinite sum has only positive terms, it must converge. It can be shown that has a nice expression in terms of the
Bernoulli numberIn mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s whenever
s is a positive even integer. With :
A rigorous proof
The following argument
provesIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
the identity ', where is the
Riemann zeta functionIn mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers...
. It is by far the most elementary proof yet available; while most proofs use results from advanced mathematics, such as Fourier analysis,
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
, and
multivariable calculusMultivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable....
, the following does not even require single-variable
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
(although a single
limitIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Informally, a function assigns an output f to every input x. The function has a limit L at an input p if f is "close" to L whenever x is...
is taken at the end).
History of this proof
The proof goes back to
Augustin Louis CauchyAugustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation...
(Cours d'Analyse, 1821, Note VIII). In 1954 this proof appeared in the book of Akiva and
Isaak YaglomIsaak Moiseevich Yaglom was a Soviet mathematician and author of popular mathematics books.Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan....
"Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal
Eureka, attributed to John Scholes, but Scholes claims he learned the proof from
Peter Swinnerton-DyerSir Henry Peter Francis Swinnerton-Dyer, 16th Baronet KBE FRS , commonly known as Peter Swinnerton-Dyer, is an English mathematician specialising in number theory at University of Cambridge. As a mathematician he is best known for his part in the Birch and Swinnerton-Dyer conjecture relating...
, and in any case he maintains the proof was "common knowledge at
CambridgeThe University of Cambridge , located in the City of Cambridge, Cambridgeshire, United Kingdom, is the second oldest university in the English-speaking world and the fourth oldest in Europe...
in the late 1960s".
What you need to know
To understand the proof, you will need to understand the following facts:
- De Moivre's formula
In mathematics, De Moivre's formula, named after Abraham de Moivre, states that for any complex number x and any integer n it holds that...
states that for any real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
x and any integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
n,
- Proof: This can be proved from Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function...
; see the article for more details.
- The binomial theorem
In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power
n into a sum involving terms of the form ax
by
c, where the coefficient of each term is a positive...
, which states that for any complex numberA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
s x and y and any nonnegative integer n,
- where the binomial coefficient
In mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial power n....
s can be expressed using the factorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
by
- Proof: See the proof in the article about the binomial theorem, it uses mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite...
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). Adding (sin2 x)(sin2 y) to 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 one-to-one on the interval (0, π/2), so that x = y.
- If p(t) = amtm + am − 1tm − 1 + ... + a1t + a0, where am ≠ 0, is a polynomial of degree m with distinct real roots t1, t2, ... , tm, then p(t) = am(t − t1)(t − t2) ... (t − tm) and t1 + t2 + ... + tm = −am−1/am.
- Proof: The product representation of the polynomial is a consequence of the factor theorem
In algebra, the factor theorem is a theorem for finding out the factors of a polynomial...
, which in turn follows from polynomial long divisionIn algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division...
. The formula for the sum of the roots is a special case of Viète's formulasIn mathematics, more specifically in algebra, Viète's Laws, named after François Viète, are formulas which relate the coefficients of a polynomial to signed sums and products of its roots.-The Laws:Any general polynomial of degree n ≥ 1,...
and follows by expanding the product representation of the polynomial and comparing the coefficient of tm-1.
- 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:
-
-
- To see that 0 < sin x < x, observe that in the picture, sin θ is the length of the line AC, and θ is the length of the circular arc AD.
- To see that x < tan x, observe that the area of the triangle OAE is tan(θ)/2, the area of the sector OAD is θ/2, and that the sector is contained within the triangle.
- 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 ≠ 0; then the limit of the sequence (am + b)/(am + c) tends to 1 as m approaches infinity.
- Proof: Divide each term by m to 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
In calculus, the squeeze theorem is a theorem regarding the limit of a function....
, 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.
The proof
The main idea behind the proof is to bound the partial sums
between two expressions, each of which will tend to π
2/6 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 odd 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, fix a positive integer
m, set
n = 2
m + 1 and consider
xr =
r π/(2
m + 1) for
r = 1, 2, ...,
m. Then
nxr is a multiple of π and therefore a zero of the sine function, and so
for every
r = 1, 2, ...,
m. The values
x1, ...,
xm are distinct numbers in the interval (0, π/2). Since the function cot
2 x is one-to-one on this interval, the numbers
tr = cot
2 xr are distinct for
r = 1, 2, ...,
m. By the above equation, these
m numbers are the roots of the
mth degree polynomial
By
Viète's formulasIn mathematics, more specifically in algebra, Viète's Laws, named after François Viète, are formulas which relate the coefficients of a polynomial to signed sums and products of its roots.-The Laws:Any general polynomial of degree n ≥ 1,...
we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that
Substituting the identity csc
2 x = cot
2 x + 1, we have
Now consider the inequality cot
2 x < 1/
x2 < csc
2 x. If we add up all these inequalities for each of the numbers
xr =
r π/(2
m + 1), and if we use the two identities above, we get
Multiplying through by (π/(2
m + 1))
2, this becomes
As
m approaches infinity, the left and right hand expressions each approach π
2/6, so by the
squeeze theoremIn calculus, the squeeze theorem is a theorem regarding the limit of a function....
,
and this completes the proof.
A slicker proof from Fourier series
Let over the interval . The
Fourier seriesIn mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines . The study of Fourier series is a branch of Fourier analysis...
for this function (worked out in that article) is
Then, using
Parseval's identityIn mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is thePythagorean theorem for inner-product spaces....
(with ) we have that
,
where
for , and .
Thus,
and
Therefore,
as required.
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