Basel Problem
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The Basel problem is a problem in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
with relevance to
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, concerning an infinite sum of inverse squares. It was first posed by
Pietro Mengoli Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647. He remained as professor there ...
in 1650 and solved by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in 1734, and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences''. Since the problem had withstood the attacks of the leading
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
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 " die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is seminal9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der K ...
", in which he defined his
zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * ...
and proved its basic properties. The problem is named after
Basel , french: link=no, Bâlois(e), it, Basilese , neighboring_municipalities= Allschwil (BL), Hégenheim (FR-68), Binningen (BL), Birsfelden (BL), Bottmingen (BL), Huningue (FR-68), Münchenstein (BL), Muttenz (BL), Reinach (BL), Riehen (BS ...
, hometown of Euler as well as of the
Bernoulli family The Bernoulli family () of Basel was a patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the early modern period. ...
who unsuccessfully attacked the problem. The Basel problem asks for the precise
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
of the reciprocals of the
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, i.e. the precise sum of the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
: \sum_^\infty \frac = \frac + \frac + \frac + \cdots. The sum of 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 \pi^2/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct. He produced a truly rigorous proof in 1741. The solution to this problem can be used to estimate the probability that two large
random number In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness. Algorithms and implementations A 1964-developed algorithm is popularly known as ''the Knuth s ...
s are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. Two random integers in the range from 1 to n, in the limit as n goes to infinity, are relatively prime with a probability that approaches 6/\pi^2, the reciprocal of the solution to the Basel problem.


Euler's approach

Euler's original derivation of the value \pi^2/6 essentially extended observations about finite
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification (100 years later,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
proved that Euler's representation of the sine function as an infinite product is valid, by the Weierstrass factorization theorem), 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion of the sine function \sin x = x - \frac + \frac - \frac + \cdots Dividing through by x gives \frac = 1 - \frac + \frac - \frac + \cdots . The Weierstrass factorization theorem shows that the left-hand side is the product of linear factors given by its roots, just as for finite polynomials. Euler assumed this as a
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
for expanding an infinite degree
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
in terms of its roots, but in fact is not always true for general P(x). This factorization expands the equation into: \begin \frac &= \left(1 - \frac\right)\left(1 + \frac\right)\left(1 - \frac\right)\left(1 + \frac\right)\left(1 - \frac\right)\left(1 + \frac\right) \cdots \\ &= \left(1 - \frac\right)\left(1 - \frac\right)\left(1 - \frac\right) \cdots \end If we formally multiply out this product and collect all the terms (we are allowed to do so because of
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynom ...
), we see by induction that the coefficient of is -\left(\frac + \frac + \frac + \cdots \right) = -\frac\sum_^\frac. But from the original infinite series expansion of , the coefficient of is . These two coefficients must be equal; thus, -\frac = -\frac\sum_^\frac. Multiplying both sides of this equation by −2 gives the sum of the reciprocals of the positive square integers. \sum_^\frac = \frac. This method of calculating \zeta(2) is detailed in expository fashion most notably in Havil's ''Gamma'' book which details many
zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * ...
and
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
-related series and integrals, as well as a historical perspective, related to the Euler gamma constant.


Generalizations of Euler's method using elementary symmetric polynomials

Using formulae obtained from
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s, this same approach can be used to enumerate formulae for the even-indexed even zeta constants which have the following known formula expanded by the
Bernoulli numbers In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
: \zeta(2n) = \frac B_. For example, let the partial product for \sin(x) expanded as above be defined by \frac := \prod\limits_^n \left(1 - \frac\right). Then using known formulas for elementary symmetric polynomials (a.k.a., Newton's formulas expanded in terms of power sum identities), we can see (for example) that \begin \left ^4\right\frac & = \frac\left(\left(H_n^\right)^2 - H_n^\right) \qquad \xrightarrow \qquad \frac\left(\zeta(2)^2-\zeta(4)\right) \\ pt& \qquad \implies \zeta(4) = \frac = -2\pi^4 \cdot ^4\frac +\frac \\ pt\left ^6\right\frac & = -\frac\left(\left(H_n^\right)^3 - 2H_n^ H_n^ + 2H_n^\right) \qquad \xrightarrow \qquad \frac\left(\zeta(2)^3-3\zeta(2)\zeta(4) + 2\zeta(6)\right) \\ pt& \qquad \implies \zeta(6) = \frac = -3 \cdot \pi^6 ^6\frac - \frac \frac \frac + \frac, \end and so on for subsequent coefficients of ^\frac. There are other forms of Newton's identities expressing the (finite) power sums H_n^ in terms of the
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s, e_i \equiv e_i\left(-\frac, -\frac, -\frac, -\frac, \ldots\right), but we can go a more direct route to expressing non-recursive formulas for \zeta(2k) using the method of
elementary symmetric polynomial In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
s. Namely, we have a recurrence relation between the elementary symmetric polynomials and the power sum polynomials given as on this page by (-1)^k e_k(x_1,\ldots,x_n) = \sum_^k (-1)^ p_j(x_1,\ldots,x_n)e_(x_1,\ldots,x_n), which in our situation equates to the limiting recurrence relation (or
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
convolution, or
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
) expanded as \frac\cdot \frac = - ^\frac \times \sum_ \zeta(2i) x^i. Then by differentiation and rearrangement of the terms in the previous equation, we obtain that \zeta(2k) = ^frac\left(1-\pi x\cot(\pi x)\right).


Consequences of Euler's proof

By the above results, we can conclude that \zeta(2k) is ''always'' a
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
multiple of \pi^. In particular, since \pi and integer powers of it are transcendental, we can conclude at this point that \zeta(2k) is
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, and more precisely, transcendental for all k \geq 1. By contrast, the properties of the odd-indexed zeta constants, including
Apéry's constant In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number : \begin \zeta(3) &= \sum_^\infty \frac \\ &= \lim_ \left(\frac + \frac + \cdots + \frac\right), \end ...
\zeta(3), are almost completely unknown.


The Riemann zeta function

The
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is one of the most significant functions in mathematics because of its relationship to the distribution of the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. The zeta function is defined for any
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
with real part greater than 1 by the following formula: \zeta(s) = \sum_^\infty \frac. Taking , we see that is equal to the sum of the reciprocals of the squares of all positive integers: \zeta(2) = \sum_^\infty \frac = \frac + \frac + \frac + \frac + \cdots = \frac \approx 1.644934. Convergence can be proven by the
integral test In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
, or by the following inequality: \begin \sum_^N \frac & < 1 + \sum_^N \frac \\ & = 1 + \sum_^N \left( \frac - \frac \right) \\ & = 1 + 1 - \frac \;\; 2. \end This gives us the
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an eleme ...
2, and because the infinite sum contains no negative terms, it must converge to a value strictly between 0 and 2. It can be shown that has a simple expression in terms of the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s whenever is a positive even integer. With : \zeta(2n) = \frac.


A proof using Euler's formula and L'Hôpital's rule

The normalized sinc function \text(x)=\frac has a Weierstrass factorization representation as an infinite product: \frac = \prod_^\infty \left(1-\frac\right). The infinite product is analytic, so taking the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of both sides and differentiating yields \frac-\frac=-\sum_^\infty \frac (by
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
, the interchange of the derivative and infinite series is permissible). After dividing the equation by 2x and regrouping one gets \frac-\frac=\sum_^\infty \frac. We make a change of variables (x=-it): -\frac+\frac=\sum_^\infty \frac.
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
can be used to deduce that \frac=\frac\frac=\frac+\frac. or using
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
: \frac=\frac=\frac\coth(\pi t). Then \sum_^\infty \frac=\frac=-\frac + \frac \coth(\pi t). Now we take the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
as t approaches zero and use
L'Hôpital's rule In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
thrice. By
Tannery's theorem In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery. Statement Let S_n = \sum_^\infty a_k(n) and suppose that \lim_ a_k ...
applied to \lim_\sum_^\infty 1/(n^2+1/t^2), we can interchange the limit and infinite series so that \lim_\sum_^\infty 1/(n^2+t^2)=\sum_^\infty 1/n^2 and by L'Hôpital's rule \begin\sum_^\infty \frac&=\lim_\frac\frac\\ pt&=\lim_\frac\\ pt&=\lim_\frac\\ pt&=\frac.\end


A proof using Fourier series

Use
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
(applied to the function ) to obtain \sum_^\infty , c_n, ^2 = \frac\int_^\pi x^2 \, dx, where \begin c_n &= \frac\int_^\pi x e^ \, dx \\ pt &= \frac i \\ pt &= \frac i \\ pt &= \frac i \end for , and . Thus, , c_n, ^2 = \begin \dfrac, & \text n \neq 0, \\ 0, & \text n = 0, \end and \sum_^\infty , c_n, ^2 = 2\sum_^\infty \frac = \frac \int_^\pi x^2 \, dx. Therefore, \sum_^\infty \frac = \frac\int_^\pi x^2 \, dx = \frac as required.


Another proof using Parseval's identity

Given a complete orthonormal basis in the space L^2_(0, 1) of L2
periodic function A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
s over (0, 1) (i.e., the subspace of square-integrable functions which are also periodic), denoted by \_^,
Parseval's identity In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is a generalized Pythagorean theorem for inner-product spaces (which ...
tells us that \, x\, ^2 = \sum_^ , \langle e_i, x\rangle, ^2, where \, x\, := \sqrt is defined in terms of the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on this Hilbert space given by \langle f, g\rangle = \int_0^1 f(x) \overline \, dx,\ f,g \in L^2_(0, 1). We can consider the
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
on this space defined by e_k \equiv e_k(\vartheta) := \exp(2\pi\imath k \vartheta) such that \langle e_k,e_j\rangle = \int_0^1 e^ \, d\vartheta = \delta_. Then if we take f(\vartheta) := \vartheta, we can compute both that \begin \, f\, ^2 & = \int_0^1 \vartheta^2 \, d\vartheta = \frac \\ \langle f, e_k\rangle & = \int_0^1 \vartheta e^ \, d\vartheta = \Biggl\}, where v_n = 2n-1 \mapsto \ and \widetilde_n = 11n^2-11n+3 \mapsto \.


See also

*
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
*
Apéry's constant In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number : \begin \zeta(3) &= \sum_^\infty \frac \\ &= \lim_ \left(\frac + \frac + \cdots + \frac\right), \end ...
*
List of sums of reciprocals In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many n ...


References

* . * . * . * * .


Notes


External links


An infinite series of surprises
by C. J. Sangwin
From ''ζ''(2) to Π. The Proof.
step-by-step proof * , English translation with notes of Euler's paper by Lucas Willis and Thomas J. Osler * * * (fourteen proofs)
Visualization of Euler's factorization of the sine function
* ** {{YouTube, d-o3eB9sfls, Why is pi here? And why is it squared? A geometric answer to the Basel problem (animated proof based on the above) Articles containing proofs Mathematical problems Number theory Pi algorithms Squares in number theory Zeta and L-functions