Basel problem
Encyclopedia
The Basel problem is a famous problem in mathematical analysis
with relevance to number theory
, first posed by Pietro Mengoli
in 1644 and solved by Leonhard Euler
in 1735. Since the problem had withstood the attacks of the leading mathematician
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 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.
The Basel problem asks for the precise summation
of the reciprocals
of the squares
of the natural number
s, i.e. the precise sum of the infinite series
:
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
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.
is clever and original. He essentially extended observations about finite polynomial
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 series
expansion of the sine function
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 (normalized) 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 are allowed to do so because of Newton's identities
), 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.
is one of the most important functions in mathematics, because of its relationship to the distribution of the prime number
s. The function is defined for any complex number
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 number
s whenever s is a positive even integer. With
:
over the interval x ∈ (–,). The Fourier series
for this function (worked out in that article) is
Then, using Parseval's identity
(with
) we have that
,
where
for n ≠ 0, and a0 = 0. Thus,
for n ≠ 0 and
Therefore,
as required.
, and multivariable calculus
, the following does not even require single-variable calculus
(although a single limit
is taken at the end).
(Cours d'Analyse, 1821, Note VIII). In 1954 this proof appeared in the book of Akiva
and Isaak Yaglom
"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-Dyer
, and in any case he maintains the proof was "common knowledge at Cambridge
in the late 1960s".
between two expressions, each of which will tend to 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
be a real number with 
, 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
, set 
and consider 
for 
. Then 
is a multiple of 
and therefore a zero of the sine function, and so
for every
. The values 
are distinct numbers in the interval (0, /2). Since the function 
is one-to-one on this interval, the numbers 
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 formulas 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
, we have
Now consider the inequality
. If we add up all these inequalities for each of the numbers 
, and if we use the two identities above, we get
Multiplying through by (/(2m + 1))2, this becomes
As m approaches infinity, the left and right hand expressions each approach
, so by the squeeze theorem
,
and this completes the proof.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
with relevance to number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, first posed by Pietro Mengoli
Pietro Mengoli
Pietro 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 Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
in 1735. Since the problem had withstood the attacks of the leading mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
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 Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude
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 Akademie der Wissenschaften zu Berlin.Although it is the only paper he ever published on number theory, it...
, in which he defined his zeta function and proved its basic properties. The problem is named after Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...
, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.
The Basel problem asks for the precise summation
Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed may be integers, rational numbers,...
of the reciprocals
Multiplicative inverse
In 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 a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of the squares
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
of the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, i.e. the precise sum of the infinite series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
:
The series is approximately equal to 1.644934 . The Basel problem asks for the exact sum of this series (in closed form
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
), as well as a proof
Mathematical proof
In 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 polynomialPolynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer 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 series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
expansion of the sine function
Trigonometric function
In 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 (normalized) 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 are allowed to do so because of Newton's identities
Newton's identities
In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials...
), 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 function is one of the most important functions in mathematics, because of its relationship to the distribution of the prime numberPrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. The function is defined for any complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 numberBernoulli number
In mathematics, the Bernoulli numbers Bn 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 using Fourier series
Let over the interval x ∈ (–,). The Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
for this function (worked out in that article) is
Then, using Parseval's identity
Parseval's identity
In 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 n ≠ 0, and a0 = 0. Thus,
for n ≠ 0 and
Therefore,
as required.
A rigorous elementary proof
This is by far the most elementary well-known proof; while most proofs use results from advanced mathematics, such as Fourier analysis, complex analysisComplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, and multivariable calculus
Multivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
, the following does not even require single-variable calculus
Calculus
Calculus is a branch of 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 theorem...
(although a single limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
is taken at the end).
History of this proof
The proof goes back to Augustin Louis CauchyAugustin Louis Cauchy
Baron Augustin-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, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
(Cours d'Analyse, 1821, Note VIII). In 1954 this proof appeared in the book of Akiva
Akiva Yaglom
Akiva Moiseevich Yaglom was a Soviet and Russian physicist, mathematician, statistician, and meteorologist. He was known for his contribution to the statistical theory of turbulence and theory of random processes. Yaglom spent most of his career in Russia working in various institutions, including...
and Isaak Yaglom
Isaak Yaglom
Isaak Moiseevich Yaglom was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom.Yaglom received a Ph.D. from Moscow State University in 1945 as student of Veniamin Kagan. As the author of several books, translated into English, that have become academic...
"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-Dyer
Peter Swinnerton-Dyer
Sir 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...
, and in any case he maintains the proof was "common knowledge at Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
in the late 1960s".
The proof
The main idea behind the proof is to bound the partial sumsbetween two expressions, each of which will tend to 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
De Moivre's formula
In mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
, and we now turn to establishing these identities.
Let
be a real number with , and let n be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we haveFrom the binomial theorem
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 axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...
, we have
Combining the two equations and equating imaginary parts gives the identity
We take this identity, fix a positive integer
, set and consider for . Then is a multiple of and therefore a zero of the sine function, and sofor every
. The values are distinct numbers in the interval (0, /2). Since the function is one-to-one on this interval, the numbers are distinct for r = 1, 2, ..., m. By the above equation, these m numbers are the roots of the mth degree polynomialBy Viète's formulas 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
, we haveNow consider the inequality
. If we add up all these inequalities for each of the numbers , and if we use the two identities above, we getMultiplying through by (/(2m + 1))2, this becomes
As m approaches infinity, the left and right hand expressions each approach
, so by the squeeze theoremSqueeze theorem
In calculus, the squeeze theorem is a theorem regarding the limit of a function....
,
and this completes the proof.