## Table of Contents

103 relations: Adele ring, Akiva Yaglom, Algebraic group, Analytic function, Antiderivative, Apéry's constant, Augustin-Louis Cauchy, Bailey–Borwein–Plouffe formula, Basel, Bernhard Riemann, Bernoulli family, Bernoulli number, Binomial theorem, Calculus, Cauchy product, Closed-form expression, Compact space, Complex analysis, Complex number, Congruence subgroup, Constant of integration, Continued fraction, Coprime integers, De Moivre's formula, Elementary symmetric polynomial, Euler product, Euler's constant, Euler's formula, Factorization, Fourier analysis, Generating function, Haar measure, Harmonic number, Heuristic, Hilbert space, Hyperbolic functions, Injective function, Inner product space, Integration by parts, Integration by substitution, Interchange of limiting operations, Irrationality, Isaak Yaglom, Karl Weierstrass, L'Hôpital's rule, Leibniz integral rule, Leonhard Euler, Limit (mathematics), Limit of a function, List of sums of reciprocals, ... Expand index (53 more) »

- Pi algorithms
- Squares in number theory

## Adele ring

In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory.

See Basel problem and Adele ring

## Akiva Yaglom

Akiva Moiseevich Yaglom (Аки́ва Моисе́евич Ягло́м; 6 March 1921 – 13 December 2007) was a Soviet and Russian physicist, mathematician, statistician, and meteorologist.

See Basel problem and Akiva Yaglom

## Algebraic group

In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety.

See Basel problem and Algebraic group

## Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

See Basel problem and Analytic function

## Antiderivative

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function.

See Basel problem and Antiderivative

## Apéry's constant

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. Basel problem and Apéry's constant are zeta and L-functions.

See Basel problem and Apéry's constant

## Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy (France:, ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist.

See Basel problem and Augustin-Louis Cauchy

## Bailey–Borwein–Plouffe formula

The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for pi. Basel problem and Bailey–Borwein–Plouffe formula are pi algorithms.

See Basel problem and Bailey–Borwein–Plouffe formula

## Basel

Basel, also known as Basle,Bâle; Basilea; Basileia; other Basilea.

## Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry.

See Basel problem and Bernhard Riemann

## 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.

See Basel problem and Bernoulli family

## Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. Basel problem and Bernoulli number are number theory.

See Basel problem and Bernoulli number

## Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

See Basel problem and Binomial theorem

## Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

See Basel problem and Calculus

## Cauchy product

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.

See Basel problem and Cauchy product

## Closed-form expression

In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (and integer powers) and function composition.

See Basel problem and Closed-form expression

## Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.

See Basel problem and Compact space

## Complex 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.

See Basel problem and Complex analysis

## 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^.

See Basel problem and Complex number

## Congruence subgroup

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.

See Basel problem and Congruence subgroup

## Constant of integration

In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected domain, is only defined up to an additive constant.

See Basel problem and Constant of integration

## Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

See Basel problem and Continued fraction

## Coprime integers

In number theory, 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. Basel problem and coprime integers are number theory.

See Basel problem and Coprime integers

## De Moivre's formula

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that where is the imaginary unit.

See Basel problem and De Moivre's formula

## 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 symmetric polynomials.

See Basel problem and Elementary symmetric polynomial

## Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. Basel problem and Euler product are zeta and L-functions.

See Basel problem and Euler product

## Euler's constant

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by: \begin \gamma &.

See Basel problem and Euler's constant

## 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.

See Basel problem and Euler's formula

## Factorization

In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

See Basel problem and Factorization

## Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

See Basel problem and Fourier analysis

## Generating function

In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.

See Basel problem and Generating function

## Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

See Basel problem and Haar measure

## Harmonic number

In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n. Basel problem and harmonic number are number theory.

See Basel problem and Harmonic number

## Heuristic

A heuristic or heuristic technique (problem solving, mental shortcut, rule of thumb) is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless "good enough" as an approximation or attribute substitution.

See Basel problem and Heuristic

## Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

See Basel problem and Hilbert space

## Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.

See Basel problem and Hyperbolic functions

## Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.

See Basel problem and Injective function

## Inner product space

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.

See Basel problem and Inner product space

## Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative.

See Basel problem and Integration by parts

## Integration by substitution

In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives.

See Basel problem and Integration by substitution

## Interchange of limiting operations

In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order.

See Basel problem and Interchange of limiting operations

## Irrationality

Irrationality is cognition, thinking, talking, or acting without rationality.

See Basel problem and Irrationality

## Isaak Yaglom

Isaak Moiseevich Yaglom (Исаа́к Моисе́евич Ягло́м; 6 March 1921 – 17 April 1988) was a Soviet mathematician and author of popular mathematics books, some with his twin Akiva Yaglom.

See Basel problem and Isaak Yaglom

## Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

See Basel problem and Karl Weierstrass

## L'Hôpital's rule

L'Hôpital's rule or L'Hospital's rule, also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives.

See Basel problem and L'Hôpital's rule

## Leibniz integral rule

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty and the integrands are functions dependent on x, the derivative of this integral is expressible as \begin & \frac \left (\int_^ f(x,t)\,dt \right) \\ &.

See Basel problem and Leibniz integral rule

## 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 many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

See Basel problem and Leonhard Euler

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value.

See Basel problem and Limit (mathematics)

## Limit of a function

Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1.

See Basel problem and Limit of a function

## 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.

See Basel problem and List of sums of reciprocals

## List of trigonometric identities

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined.

See Basel problem and List of trigonometric identities

## List of zeta functions

In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function Zeta functions include. Basel problem and List of zeta functions are zeta and L-functions.

See Basel problem and List of zeta functions

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

See Basel problem and Logarithm

## Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

See Basel problem and Mathematical analysis

## Mathematical Association of America

The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level.

See Basel problem and Mathematical Association of America

## Mathematical induction

Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold.

See Basel problem and Mathematical induction

## Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

See Basel problem and Mathematical proof

## Mathematician

A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.

See Basel problem and Mathematician

## Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Basel problem and Mathematics

## Mercator series

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: In summation notation, The series converges to the natural logarithm (shifted by 1) whenever -1.

See Basel problem and Mercator series

## 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.

See Basel problem and Multiplicative inverse

## Multivariable calculus

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables (multivariate), rather than just one.

See Basel problem and Multivariable calculus

## Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to.

See Basel problem and Natural logarithm

## Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0. Basel problem and natural number are number theory.

See Basel problem and Natural number

## 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.

See Basel problem and Newton's identities

## 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 functions.

See Basel problem and Number theory

## 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 a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.

See Basel problem and On the Number of Primes Less Than a Given Magnitude

## 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.

See Basel problem and Orthonormal basis

## P-adic number

In number theory, given a prime number, the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. Basel problem and p-adic number are number theory.

See Basel problem and P-adic number

## 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.

See Basel problem and Parseval's identity

## Partial fraction decomposition

In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

See Basel problem and Partial fraction decomposition

## Particular values of the Riemann zeta function

In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. Basel problem and Particular values of the Riemann zeta function are zeta and L-functions.

See Basel problem and Particular values of the Riemann zeta function

## Periodic function

A periodic function or cyclic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods.

See Basel problem and Periodic function

## Peter Swinnerton-Dyer

Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet, (2 August 1927 – 26 December 2018) was an English mathematician specialising in number theory at the University of Cambridge.

See Basel problem and Peter Swinnerton-Dyer

## 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.

See Basel problem and Pietro Mengoli

## Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

See Basel problem and Polynomial

## Power sum symmetric polynomial

In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients.

See Basel problem and Power sum symmetric polynomial

## 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.

See Basel problem and Prime number

## Random number

A random number is generated by a random (stochastic) process such as throwing Dice.

See Basel problem and Random number

## Rationality

Rationality is the quality of being guided by or based on reason.

See Basel problem and Rationality

## Recurrence relation

In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms.

See Basel problem and Recurrence relation

## 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). Basel problem and Riemann zeta function are zeta and L-functions.

See Basel problem and Riemann zeta function

## Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

See Basel problem and Series (mathematics)

## Sinc function

In mathematics, physics and engineering, the sinc function, denoted by, has two forms, normalized and unnormalized.

See Basel problem and Sinc function

## Square number

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. Basel problem and square number are number theory and squares in number theory.

See Basel problem and Square number

## Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

See Basel problem and Square-integrable function

## Squeeze theorem

In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is bounded between two other functions.

See Basel problem and Squeeze theorem

## Summation

In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.

See Basel problem and Summation

## Sums of powers

In mathematics and statistics, sums of powers occur in a number of contexts. Basel problem and sums of powers are number theory and squares in number theory.

See Basel problem and Sums of powers

## Tamagawa number

In mathematics, the Tamagawa number \tau(G) of a semisimple algebraic group defined over a global field is the measure of G(\mathbb)/G(k), where \mathbb is the adele ring of.

See Basel problem and Tamagawa number

## Tannery's theorem

In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations.

See Basel problem and Tannery's theorem

## 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.

See Basel problem and Taylor series

## The Mathematical Intelligencer

The Mathematical Intelligencer is a mathematical journal published by Springer Science+Business Media that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals.

See Basel problem and The Mathematical Intelligencer

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients.

See Basel problem and Transcendental number

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

See Basel problem and Trigonometric functions

## Trigonometric substitution

In mathematics, a trigonometric substitution replaces a trigonometric function for another expression.

See Basel problem and Trigonometric substitution

## University of Cambridge

The University of Cambridge is a public collegiate research university in Cambridge, England.

See Basel problem and University of Cambridge

## Upper and lower bounds

In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of.

See Basel problem and Upper and lower bounds

## Vieta's formulas

In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

See Basel problem and Vieta's formulas

## Vladimir Platonov

Vladimir Petrovich Platonov (Уладзімір Пятровіч Платонаў, Uladzimir Piatrovic Platonau; Влади́мир Петро́вич Плато́нов; born 1 December 1939, Stayki village, Vitebsk Region, Byelorussian SSR) is a Soviet, Belarusian and Russian mathematician.

See Basel problem and Vladimir Platonov

## Volume form

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension.

See Basel problem and Volume form

## Weierstrass factorization theorem

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes.

See Basel problem and Weierstrass factorization theorem

## Weil's conjecture on Tamagawa numbers

In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number \tau(G) of a simply connected simple algebraic group defined over a number field is 1.

See Basel problem and Weil's conjecture on Tamagawa numbers

## See also

### Pi algorithms

- Approximations of π
- Bailey–Borwein–Plouffe formula
- Basel problem
- Bellard's formula
- Borwein's algorithm
- Chronology of computation of π
- Chudnovsky algorithm
- FEE method
- Gauss–Legendre algorithm
- Leibniz formula for π
- List of formulae involving π
- Liu Hui's π algorithm
- Machin-like formula
- Ramanujan–Sato series
- Viète's formula
- Wallis product
- Zhao Youqin's π algorithm

### Squares in number theory

- Büchi's problem
- Basel problem
- Brahmagupta–Fibonacci identity
- Brocard's conjecture
- Congruence of squares
- Congruum
- Degen's eight-square identity
- Dixon's factorization method
- Euler's criterion
- Euler's four-square identity
- Fermat's right triangle theorem
- Fermat's theorem on sums of two squares
- Gauss's lemma (number theory)
- History of the Theory of Numbers
- Integer triangle
- Jacobi's four-square theorem
- Lagrange's four-square theorem
- Legendre's conjecture
- Legendre's three-square theorem
- Olga Taussky-Todd
- Pythagorean prime
- Pythagorean quadruple
- Pythagorean triple
- Quadratic form
- Ramanujan's sum
- Ramanujan's ternary quadratic form
- Square (algebra)
- Square number
- Sum of squares function
- Sum of two squares theorem
- Sums of powers
- Sun Zhiwei
- The Book of Squares
- Triangular number
- Unit square
- Waring's problem
- Zolotarev's lemma

## References

Also known as 1 + 1/4 + 1/9 + 1/16 + · · ·, Basel series, Basel sum, Basler problem, Evaluation of z(2), Evaluation of ζ(2), Riemann zeta function zeta(2), Series of reciprocal squares, Sum of the reciprocals of the square numbers, Sum of the reciprocals of the squares, Zeta(2), Π^2/6.