{"history":[],"article":{"id":"d16baad4-65bf-41dd-9854-51ad155592a8","title":"Basel problem","slug":"basel-problem","content":":::summary\n:::\n\n# \nThe **Basel problem** is a problem in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) with relevance to [number theory](https://bliptext.com/articles/number-theory), concerning an infinite sum of inverse squares.\nIt was first posed by [Pietro Mengoli](https://bliptext.com/articles/pietro-mengoli) in 1650 and solved by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences.\nSince the problem had withstood the attacks of the leading [mathematicians](https://bliptext.com/articles/mathematician) of the day, Euler's solution brought him immediate fame when he was twenty-eight.\nEuler generalised the problem considerably, and his ideas were taken up more than a century later by [Bernhard Riemann](https://bliptext.com/articles/bernhard-riemann) in his seminal 1859 paper \"[On the Number of Primes Less Than a Given Magnitude](https://bliptext.com/articles/on-the-number-of-primes-less-than-a-given-magnitude)\", in which he defined his [zeta function](https://bliptext.com/articles/riemann-zeta-function) and proved its basic properties.\nThe problem is named after [Basel](https://bliptext.com/articles/basel), hometown of Euler as well as of the [Bernoulli family](https://bliptext.com/articles/bernoulli-family) who unsuccessfully attacked the problem.\nThe Basel problem asks for the precise [summation](https://bliptext.com/articles/summation) of the [reciprocals](https://bliptext.com/articles/multiplicative-inverse) of the [squares](https://bliptext.com/articles/square-number) of the [natural numbers](https://bliptext.com/articles/natural-number), i.e. the precise sum of the [infinite series](https://bliptext.com/articles/series-mathematics):\nThe sum of the series is approximately equal to 1.644934.\nThe Basel problem asks for the exact sum of this series (in [closed form](https://bliptext.com/articles/closed-form-expression)), as well as a [proof](https://bliptext.com/articles/mathematical-proof) that this sum is correct.\nEuler found the exact sum to be \\pi^2/6 and announced this discovery in 1735.\nHis arguments were based on manipulations that were not justified at the time, although he was later proven correct.\nHe produced an accepted proof in 1741.\nThe solution to this problem can be used to estimate the probability that two large [random numbers](https://bliptext.com/articles/random-number) are [coprime](https://bliptext.com/articles/coprime).\nTwo 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.\n\n# Euler's approach\nEuler's original derivation of the value \\pi^2/6 essentially extended observations about finite [polynomials](https://bliptext.com/articles/polynomial) and assumed that these same properties hold true for infinite series.\nOf course, Euler's original reasoning requires justification (100 years later, [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) proved that Euler's representation of the sine function as an infinite product is valid, by the [Weierstrass factorization theorem](https://bliptext.com/articles/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.\nThe agreement he observed gave him sufficient confidence to announce his result to the mathematical community.\nTo follow Euler's argument, recall the [Taylor series](https://bliptext.com/articles/taylor-series) expansion of the [sine function](https://bliptext.com/articles/trigonometric-function)\nDividing through by x gives\nThe [Weierstrass factorization theorem](https://bliptext.com/articles/weierstrass-factorization-theorem) shows that the right-hand side is the product of linear factors given by its roots, just as for finite polynomials.\nEuler assumed this as a [heuristic](https://bliptext.com/articles/heuristic) for expanding an infinite degree [polynomial](https://bliptext.com/articles/polynomial) in terms of its roots, but in fact it is not always true for general P(x).\nThis factorization expands the equation into:\nIf we formally multiply out this product and collect all the\nx2\nterms (we are allowed to do so because of [Newton's identities](https://bliptext.com/articles/newton-s-identities)), we see by induction that the\nx2\ncoefficient of\nsin x⁄x\nis\nBut from the original infinite series expansion of\nsin x⁄x\n, the coefficient of\nx2\nis\n−1⁄3! = −1⁄6\n. These two coefficients must be equal; thus,\nMultiplying both sides of this equation by −π2 gives the sum of the reciprocals of the positive square integers.\nThis method of calculating \\zeta(2) is detailed in expository fashion most notably in Havil's Gamma book which details many [zeta function](https://bliptext.com/articles/zeta-function) and [logarithm](https://bliptext.com/articles/logarithm)-related series and integrals, as well as a historical perspective, related to the [Euler gamma constant](https://bliptext.com/articles/euler-mascheroni-constant).\n\n## Generalizations of Euler's method using elementary symmetric polynomials\nUsing formulae obtained from [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomial), this same approach can be used to enumerate formulae for the even-indexed [even zeta constants](https://bliptext.com/articles/zeta-constants) which have the following known formula expanded by the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers):\nFor example, let the partial product for \\sin(x) expanded as above be defined by.\nThen using known [formulas for elementary symmetric polynomials](https://bliptext.com/articles/newton-s-identities) (a.k.a., Newton's formulas expanded in terms of [power sum](https://bliptext.com/articles/power-sum) identities), we can see (for example) that\nand so on for subsequent coefficients of.\nThere are [other forms of Newton's identities](https://bliptext.com/articles/newton-s-identities) expressing the (finite) power sums H_n^{(2k)} in terms of the [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomial), but we can go a more direct route to expressing non-recursive formulas for \\zeta(2k) using the method of [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomial).\nNamely, we have a recurrence relation between the elementary symmetric polynomials and the [power sum polynomials](https://bliptext.com/articles/power-sum-symmetric-polynomial) given as on [this page](https://bliptext.com/articles/newton-s-identities) by\nwhich in our situation equates to the limiting recurrence relation (or [generating function](https://bliptext.com/articles/generating-function) convolution, or [product](https://bliptext.com/articles/cauchy-product)) expanded as\nThen by differentiation and rearrangement of the terms in the previous equation, we obtain that\n\n## Consequences of Euler's proof\nBy the above results, we can conclude that \\zeta(2k) is always a [rational](https://bliptext.com/articles/rational) multiple of \\pi^{2k}.\nIn particular, since \\pi and integer powers of it are [transcendental](https://bliptext.com/articles/[transcendental](https://bliptext.com/articles/transcendental-number)-number), we can conclude at this point that \\zeta(2k) is [irrational](https://bliptext.com/articles/irrational), and more precisely, [transcendental](https://bliptext.com/articles/[transcendental](https://bliptext.com/articles/transcendental-number)-number) for all k \\geq 1.\nBy contrast, the properties of the odd-indexed [zeta constants](https://bliptext.com/articles/zeta-constants), including [Apéry's constant](https://bliptext.com/articles/ap-ry-s-constant) \\zeta(3), are almost completely unknown.\n\n# The Riemann zeta function\nThe [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function)\nζ(s)\nis one of the most significant functions in mathematics because of its relationship to the distribution of the [prime numbers](https://bliptext.com/articles/prime-number).\nThe zeta function is defined for any [complex number](https://bliptext.com/articles/complex-number)\ns\nwith real part greater than 1 by the following formula:\nTaking\n, we see that\nζ(2)\nis equal to the sum of the reciprocals of the squares of all positive integers:\nConvergence can be proven by the [integral test](https://bliptext.com/articles/integral-test-for-convergence), or by the following inequality:\nThis gives us the [upper bound](https://bliptext.com/articles/upper-bound) 2, and because the infinite sum contains no negative terms, it must converge to a value strictly between 0 and 2.\nIt can be shown that\nζ(s)\nhas a simple expression in terms of the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-number) whenever\ns\nis a positive even integer.\nWith\n\n# A proof using Euler's formula and L'Hôpital's rule\nThe normalized [sinc function](https://bliptext.com/articles/sinc-function) has a [Weierstrass factorization](https://bliptext.com/articles/weierstrass-factorization-theorem) representation as an infinite product:\nThe infinite product is [analytic](https://bliptext.com/articles/analytic-function), so taking the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of both sides and differentiating yields\n(by [uniform convergence](https://bliptext.com/articles/uniform-convergence), the interchange of the derivative and infinite series is permissible).\nAfter dividing the equation by 2x and regrouping one gets\nWe make a change of variables (x=-it):\n[Euler's formula](https://bliptext.com/articles/euler-s-formula) can be used to deduce that\nor using the corresponding [hyperbolic function](https://bliptext.com/articles/hyperbolic-function):\nThen\nNow we take the [limit](https://bliptext.com/articles/limit-mathematics) as t approaches zero and use [L'Hôpital's rule](https://bliptext.com/articles/l-h-pital-s-rule) thrice.\nBy [Tannery's theorem](https://bliptext.com/articles/tannery-s-theorem) applied to, we can [interchange the limit and infinite series](https://bliptext.com/articles/interchange-of-limiting-operations) so that and by L'Hôpital's rule\n\n# A proof using Fourier series\nUse [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) (applied to the function\nf(x) = x\n) to obtain\nwhere\nfor\nn ≠ 0\n, and\n. Thus,\nand\nTherefore,\nas required.\n\n# Another proof using Parseval's identity\nGiven a [complete orthonormal basis](https://bliptext.com/articles/complete-orthonormal-basis) in the space of [L2](https://bliptext.com/articles/lp-space) [periodic functions](https://bliptext.com/articles/[periodic](https://bliptext.com/articles/periodic-function)-function) over (0, 1) (i.e., the subspace of [square-integrable functions](https://bliptext.com/articles/square-integrable-function) which are also [periodic](https://bliptext.com/articles/periodic-function)), denoted by, [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) tells us that\nwhere is defined in terms of the [inner product](https://bliptext.com/articles/inner-product) on this [Hilbert space](https://bliptext.com/articles/hilbert-space) given by\nWe can consider the [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) on this space defined by such that.\nThen if we take, we can compute both that\nby elementary [calculus](https://bliptext.com/articles/calculus) and [integration by parts](https://bliptext.com/articles/integration-by-parts), respectively.\nFinally, by [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) stated in the form above, we obtain that\n\n## Generalizations and recurrence relations\nNote that by considering higher-order powers of we can use [integration by parts](https://bliptext.com/articles/integration-by-parts) to extend this method to enumerating formulas for \\zeta(2j) when j > 1.\nIn particular, suppose we let\nso that [integration by parts](https://bliptext.com/articles/integration-by-parts) yields the [recurrence relation](https://bliptext.com/articles/recurrence-relation) that\nThen by applying [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) as we did for the first case above along with the linearity of the [inner product](https://bliptext.com/articles/inner-product) yields that\n\n# Proof using differentiation under the integral sign\nIt's possible to prove the result using elementary calculus by applying the [differentiation under the integral sign](https://bliptext.com/articles/leibniz-integral-rule) technique to an integral due to Freitas:\nWhile the [primitive function](https://bliptext.com/articles/primitive-function) of the integrand cannot be expressed in terms of elementary functions, by differentiating with respect to \\alpha we arrive at\nwhich can be integrated by [substituting](https://bliptext.com/articles/integration-by-substitution) u=e^{-x} and decomposing into [partial fractions](https://bliptext.com/articles/partial-fractions).\nIn the range the definite integral reduces to\nThe expression can be simplified using the [arctangent addition formula](https://bliptext.com/articles/inverse-trigonometric-functions) and integrated with respect to \\alpha by means of [trigonometric substitution](https://bliptext.com/articles/trigonometric-substitution), resulting in\nThe [integration constant](https://bliptext.com/articles/integration-constant) c can be determined by noticing that two distinct values of I(\\alpha) are related by\nbecause when calculating I(2) we can [factor](https://bliptext.com/articles/factorization) and express it in terms of I(0) using the [logarithm of a power identity](https://bliptext.com/articles/list-of-logarithmic-identities) and the [substitution](https://bliptext.com/articles/integration-by-substitution) u=x/2.\nThis makes it possible to determine, and it follows that\nThis final integral can be evaluated by expanding the natural logarithm into its [Taylor series](https://bliptext.com/articles/mercator-series):\nThe last two identities imply\n\n# Cauchy's proof\nWhile most proofs use results from advanced [mathematics](https://bliptext.com/articles/mathematics), such as [Fourier analysis](https://bliptext.com/articles/fourier-analysis), [complex analysis](https://bliptext.com/articles/complex-analysis), and [multivariable calculus](https://bliptext.com/articles/multivariable-[calculus](https://bliptext.com/articles/calculus)), the following does not even require single-variable [calculus](https://bliptext.com/articles/calculus) (until a single [limit](https://bliptext.com/articles/limit-of-a-function) is taken at the end).\nFor a proof using the residue theorem, see [here](https://bliptext.com/articles/residue-theorem).\n\n## History of this proof\nThe proof goes back to [Augustin Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) (Cours d'Analyse, 1821, Note VIII).\nIn 1954, this proof appeared in the book of [Akiva](https://bliptext.com/articles/akiva-yaglom) and [Isaak Yaglom](https://bliptext.com/articles/isaak-yaglom) \"Nonelementary Problems in an Elementary Exposition\".\nLater, in 1982, it appeared in the journal Eureka, attributed to John Scholes, but Scholes claims he learned the proof from [Peter Swinnerton-Dyer](https://bliptext.com/articles/peter-swinnerton-dyer), and in any case he maintains the proof was \"common knowledge at [Cambridge](https://bliptext.com/articles/university-of-cambridge) in the late 1960s\".\n\n## The proof\n[[File:limit circle FbN.jpeg|thumb|The inequality is shown pictorially for any . The three terms are the areas of the triangle OAC, circle section OAB, and the triangle OAB.\nTaking reciprocals and squaring gives\n.]] The main idea behind the proof is to bound the partial (finite) sums\nbetween two expressions, each of which will tend to π2⁄6 as\nm\napproaches infinity.\nThe two expressions are derived from identities involving the [cotangent](https://bliptext.com/articles/cotangent) and [cosecant](https://bliptext.com/articles/cosecant) functions.\nThese identities are in turn derived from [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula), and we now turn to establishing these identities.\nLet\nx\nbe a real number with\n0 < x < π⁄2\n, and let\nn\nbe a positive odd integer.\nThen from de Moivre's formula and the definition of the cotangent function, we have\nFrom the [binomial theorem](https://bliptext.com/articles/binomial-theorem), we have\nCombining the two equations and equating imaginary parts gives the identity\nWe take this identity, fix a positive integer\nm\n, set\n, and consider\nfor\n. Then\nnxr\nis a multiple of π and therefore\n. So,\nfor every\n. The values\nare distinct numbers in the interval\n0 <\nxr\n< π⁄2\n. Since the function\ncot2 x\nis [one-to-one](https://bliptext.com/articles/injective-function) on this interval, the numbers\nare distinct for\n. By the above equation, these\nm\nnumbers are the roots of the\nm\nth degree polynomial\nBy [Vieta's formulas](https://bliptext.com/articles/vieta-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\nSubstituting the [identity](https://bliptext.com/articles/list-of-trigonometric-identities)\n, we have\nNow consider the inequality\ncot2 x < 1⁄x2 < csc2 x\n(illustrated geometrically above).\nIf we add up all these inequalities for each of the numbers\n, and if we use the two identities above, we get\nMultiplying through by\n( π⁄2m + 1 ) 2\n, this becomes\nAs\nm\napproaches infinity, the left and right hand expressions each approach π2⁄6, so by the [squeeze theorem](https://bliptext.com/articles/squeeze-theorem),\nand this completes the proof.\n\n# Proof assuming Weil's conjecture on Tamagawa numbers\nA proof is also possible assuming [Weil's conjecture on Tamagawa numbers](https://bliptext.com/articles/weil-s-conjecture-on-tamagawa-numbers).\nThe conjecture asserts for the case of the [algebraic group](https://bliptext.com/articles/algebraic-group) SL2(**R**) that the [Tamagawa number](https://bliptext.com/articles/tamagawa-number) of the group is one.\nThat is, the quotient of the special linear group over the rational [adeles](https://bliptext.com/articles/adele-ring) by the special linear group of the rationals (a [compact set](https://bliptext.com/articles/compact-set), because is a lattice in the [adeles](https://bliptext.com/articles/adele-ring)) has Tamagawa measure 1:\nTo determine a Tamagawa measure, the group SL_2 consists of matrices\nwith xt-yz=1.\nAn invariant [volume form](https://bliptext.com/articles/volume-form) on the group is\nThe measure of the quotient is the product of the measures of corresponding to the infinite place, and the measures of in each finite place, where \\mathbb Z_p is the [p-adic integers](https://bliptext.com/articles/p-adic-integers).\nFor the local factors,\nwhere F_p is the field with p elements, and is the [congruence subgroup](https://bliptext.com/articles/congruence-subgroup) modulo p. Since each of the coordinates x,y,z map the latter group onto and, the measure of is , where \\mu_p is the normalized [Haar measure](https://bliptext.com/articles/haar-measure) on \\mathbb Z_p.\nAlso, a standard computation shows that.\nPutting these together gives.\nAt the infinite place, an integral computation over the fundamental domain of shows that, and therefore the Weil conjecture finally gives\nOn the right-hand side, we recognize the [Euler product](https://bliptext.com/articles/euler-product) for 1/\\zeta(2), and so this gives the solution to the Basel problem.\nThis approach shows the connection between (hyperbolic) geometry and arithmetic, and can be inverted to give a proof of the Weil conjecture for the special case of SL_2, contingent on an independent proof that.\n\n# Geometric proof\nThe Basel problem can be proved with [Euclidean geometry](https://bliptext.com/articles/euclidean-geometry), using the insight that the real line can be seen as a circle of infinite radius.\nAn intuitive, if not completely rigorous sketch is given here.\nthat is,\nor\n.\nThe independence of the sisc from N can be proved easily with Euclidean geometry for the more restrictive case where N is a power of 2, i.e. N = 2^n, which still allows the limiting argument to be applied.\nThe proof proceeds by [induction](https://bliptext.com/articles/mathematical-induction) on n, and uses the [Inverse Pythagorean Theorem](https://bliptext.com/articles/inverse-pythagorean-theorem), which states that:\nwhere a and b are the cathetes and h is the height of a right triangle.\n# [The sum of inverse squares of distances of P1 and P2 from Q equals the inverse square distance from P to Q. | upload.wikimedia.org/wikipedia/commons/3/3e/Induction///step///1728668638527.jpg]\n\n# Other identities\nSee the special cases of the identities for the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) when s = 2.\nOther notably special identities and representations of this constant appear in the sections below.\n\n## Series representations\nThe following are series representations of the constant:\nThere are also [BBP-type](https://bliptext.com/articles/bailey-borwein-plouffe-formula) series expansions for\nζ(2)\n.\n\n## Integral representations\nThe following are integral representations of\n\n## Continued fractions\nIn van der Poorten's classic article chronicling [Apéry's proof of the irrationality of \\zeta(3)](https://bliptext.com/articles/ap-ry-s-constant), the author notes as \"a red herring\" the similarity of a [simple continued fraction](https://bliptext.com/articles/simple-continued-fraction) for Apery's constant, and the following one for the Basel constant:\nwhere.\nAnother continued fraction of a similar form is:\nwhere.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:47:48.051Z","updated_at":"2025-01-25T19:47:48.051Z","search_vector":"'basel':1 'problem':2","search_content":" \n \n\n# \nThe **Basel problem** is a problem in [mathematical analysis](https://bliptext.com/articles/mathematical-analysis) with relevance to [number theory](https://bliptext.com/articles/number-theory), concerning an infinite sum of inverse squares.\nIt was first posed by [Pietro Mengoli](https://bliptext.com/articles/pietro-mengoli) in 1650 and solved by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences.\nSince the problem had withstood the attacks of the leading [mathematicians](https://bliptext.com/articles/mathematician) of the day, Euler's solution brought him immediate fame when he was twenty-eight.\nEuler generalised the problem considerably, and his ideas were taken up more than a century later by [Bernhard Riemann](https://bliptext.com/articles/bernhard-riemann) in his seminal 1859 paper \"[On the Number of Primes Less Than a Given Magnitude](https://bliptext.com/articles/on-the-number-of-primes-less-than-a-given-magnitude)\", in which he defined his [zeta function](https://bliptext.com/articles/riemann-zeta-function) and proved its basic properties.\nThe problem is named after [Basel](https://bliptext.com/articles/basel), hometown of Euler as well as of the [Bernoulli family](https://bliptext.com/articles/bernoulli-family) who unsuccessfully attacked the problem.\nThe Basel problem asks for the precise [summation](https://bliptext.com/articles/summation) of the [reciprocals](https://bliptext.com/articles/multiplicative-inverse) of the [squares](https://bliptext.com/articles/square-number) of the [natural numbers](https://bliptext.com/articles/natural-number), i.e. the precise sum of the [infinite series](https://bliptext.com/articles/series-mathematics):\nThe sum of the series is approximately equal to 1.644934.\nThe Basel problem asks for the exact sum of this series (in [closed form](https://bliptext.com/articles/closed-form-expression)), as well as a [proof](https://bliptext.com/articles/mathematical-proof) that this sum is correct.\nEuler found the exact sum to be \\pi^2/6 and announced this discovery in 1735.\nHis arguments were based on manipulations that were not justified at the time, although he was later proven correct.\nHe produced an accepted proof in 1741.\nThe solution to this problem can be used to estimate the probability that two large [random numbers](https://bliptext.com/articles/random-number) are [coprime](https://bliptext.com/articles/coprime).\nTwo 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.\n\n# Euler's approach\nEuler's original derivation of the value \\pi^2/6 essentially extended observations about finite [polynomials](https://bliptext.com/articles/polynomial) and assumed that these same properties hold true for infinite series.\nOf course, Euler's original reasoning requires justification (100 years later, [Karl Weierstrass](https://bliptext.com/articles/karl-weierstrass) proved that Euler's representation of the sine function as an infinite product is valid, by the [Weierstrass factorization theorem](https://bliptext.com/articles/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.\nThe agreement he observed gave him sufficient confidence to announce his result to the mathematical community.\nTo follow Euler's argument, recall the [Taylor series](https://bliptext.com/articles/taylor-series) expansion of the [sine function](https://bliptext.com/articles/trigonometric-function)\nDividing through by x gives\nThe [Weierstrass factorization theorem](https://bliptext.com/articles/weierstrass-factorization-theorem) shows that the right-hand side is the product of linear factors given by its roots, just as for finite polynomials.\nEuler assumed this as a [heuristic](https://bliptext.com/articles/heuristic) for expanding an infinite degree [polynomial](https://bliptext.com/articles/polynomial) in terms of its roots, but in fact it is not always true for general P(x).\nThis factorization expands the equation into:\nIf we formally multiply out this product and collect all the\nx2\nterms (we are allowed to do so because of [Newton's identities](https://bliptext.com/articles/newton-s-identities)), we see by induction that the\nx2\ncoefficient of\nsin x⁄x\nis\nBut from the original infinite series expansion of\nsin x⁄x\n, the coefficient of\nx2\nis\n−1⁄3! = −1⁄6\n. These two coefficients must be equal; thus,\nMultiplying both sides of this equation by −π2 gives the sum of the reciprocals of the positive square integers.\nThis method of calculating \\zeta(2) is detailed in expository fashion most notably in Havil's Gamma book which details many [zeta function](https://bliptext.com/articles/zeta-function) and [logarithm](https://bliptext.com/articles/logarithm)-related series and integrals, as well as a historical perspective, related to the [Euler gamma constant](https://bliptext.com/articles/euler-mascheroni-constant).\n\n## Generalizations of Euler's method using elementary symmetric polynomials\nUsing formulae obtained from [elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomial), this same approach can be used to enumerate formulae for the even-indexed [even zeta constants](https://bliptext.com/articles/zeta-constants) which have the following known formula expanded by the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-numbers):\nFor example, let the partial product for \\sin(x) expanded as above be defined by.\nThen using known [formulas for elementary symmetric polynomials](https://bliptext.com/articles/newton-s-identities) (a.k.a., Newton's formulas expanded in terms of [power sum](https://bliptext.com/articles/power-sum) identities), we can see (for example) that\nand so on for subsequent coefficients of.\nThere are [other forms of Newton's identities](https://bliptext.com/articles/newton-s-identities) expressing the (finite) power sums H_n^{(2k)} in terms of the [[elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomial)](https://bliptext.com/articles/elementary-symmetric-polynomial), but we can go a more direct route to expressing non-recursive formulas for \\zeta(2k) using the method of [[elementary symmetric polynomials](https://bliptext.com/articles/elementary-symmetric-polynomial)](https://bliptext.com/articles/elementary-symmetric-polynomial).\nNamely, we have a recurrence relation between the elementary symmetric polynomials and the [power sum polynomials](https://bliptext.com/articles/power-sum-symmetric-polynomial) given as on [this page](https://bliptext.com/articles/newton-s-identities) by\nwhich in our situation equates to the limiting recurrence relation (or [generating function](https://bliptext.com/articles/generating-function) convolution, or [product](https://bliptext.com/articles/cauchy-product)) expanded as\nThen by differentiation and rearrangement of the terms in the previous equation, we obtain that\n\n## Consequences of Euler's proof\nBy the above results, we can conclude that \\zeta(2k) is always a [rational](https://bliptext.com/articles/rational) multiple of \\pi^{2k}.\nIn particular, since \\pi and integer powers of it are [[transcendental](https://bliptext.com/articles/transcendental-number)](https://bliptext.com/articles/[transcendental](https://bliptext.com/articles/transcendental-number)-number), we can conclude at this point that \\zeta(2k) is [irrational](https://bliptext.com/articles/irrational), and more precisely, [[transcendental](https://bliptext.com/articles/transcendental-number)](https://bliptext.com/articles/[transcendental](https://bliptext.com/articles/transcendental-number)-number) for all k \\geq 1.\nBy contrast, the properties of the odd-indexed [zeta constants](https://bliptext.com/articles/zeta-constants), including [Apéry's constant](https://bliptext.com/articles/ap-ry-s-constant) \\zeta(3), are almost completely unknown.\n\n# The Riemann zeta function\nThe [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function)\nζ(s)\nis one of the most significant functions in mathematics because of its relationship to the distribution of the [prime numbers](https://bliptext.com/articles/prime-number).\nThe zeta function is defined for any [complex number](https://bliptext.com/articles/complex-number)\ns\nwith real part greater than 1 by the following formula:\nTaking\n, we see that\nζ(2)\nis equal to the sum of the reciprocals of the squares of all positive integers:\nConvergence can be proven by the [integral test](https://bliptext.com/articles/integral-test-for-convergence), or by the following inequality:\nThis gives us the [upper bound](https://bliptext.com/articles/upper-bound) 2, and because the infinite sum contains no negative terms, it must converge to a value strictly between 0 and 2.\nIt can be shown that\nζ(s)\nhas a simple expression in terms of the [Bernoulli numbers](https://bliptext.com/articles/bernoulli-number) whenever\ns\nis a positive even integer.\nWith\n\n# A proof using Euler's formula and L'Hôpital's rule\nThe normalized [sinc function](https://bliptext.com/articles/sinc-function) has a [Weierstrass factorization](https://bliptext.com/articles/weierstrass-factorization-theorem) representation as an infinite product:\nThe infinite product is [analytic](https://bliptext.com/articles/analytic-function), so taking the [natural logarithm](https://bliptext.com/articles/natural-logarithm) of both sides and differentiating yields\n(by [uniform convergence](https://bliptext.com/articles/uniform-convergence), the interchange of the derivative and infinite series is permissible).\nAfter dividing the equation by 2x and regrouping one gets\nWe make a change of variables (x=-it):\n[Euler's formula](https://bliptext.com/articles/euler-s-formula) can be used to deduce that\nor using the corresponding [hyperbolic function](https://bliptext.com/articles/hyperbolic-function):\nThen\nNow we take the [limit](https://bliptext.com/articles/limit-mathematics) as t approaches zero and use [L'Hôpital's rule](https://bliptext.com/articles/l-h-pital-s-rule) thrice.\nBy [Tannery's theorem](https://bliptext.com/articles/tannery-s-theorem) applied to, we can [interchange the limit and infinite series](https://bliptext.com/articles/interchange-of-limiting-operations) so that and by L'Hôpital's rule\n\n# A proof using Fourier series\nUse [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) (applied to the function\nf(x) = x\n) to obtain\nwhere\nfor\nn ≠ 0\n, and\n. Thus,\nand\nTherefore,\nas required.\n\n# Another proof using Parseval's identity\nGiven a [complete orthonormal basis](https://bliptext.com/articles/complete-orthonormal-basis) in the space of [L2](https://bliptext.com/articles/lp-space) [[periodic](https://bliptext.com/articles/periodic-function) functions](https://bliptext.com/articles/[periodic](https://bliptext.com/articles/periodic-function)-function) over (0, 1) (i.e., the subspace of [square-integrable functions](https://bliptext.com/articles/square-integrable-function) which are also [periodic](https://bliptext.com/articles/periodic-function)), denoted by, [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) tells us that\nwhere is defined in terms of the [inner product](https://bliptext.com/articles/inner-product) on this [Hilbert space](https://bliptext.com/articles/hilbert-space) given by\nWe can consider the [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) on this space defined by such that.\nThen if we take, we can compute both that\nby elementary [calculus](https://bliptext.com/articles/calculus) and [integration by parts](https://bliptext.com/articles/integration-by-parts), respectively.\nFinally, by [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) stated in the form above, we obtain that\n\n## Generalizations and recurrence relations\nNote that by considering higher-order powers of we can use [integration by parts](https://bliptext.com/articles/integration-by-parts) to extend this method to enumerating formulas for \\zeta(2j) when j > 1.\nIn particular, suppose we let\nso that [integration by parts](https://bliptext.com/articles/integration-by-parts) yields the [recurrence relation](https://bliptext.com/articles/recurrence-relation) that\nThen by applying [Parseval's identity](https://bliptext.com/articles/parseval-s-identity) as we did for the first case above along with the linearity of the [inner product](https://bliptext.com/articles/inner-product) yields that\n\n# Proof using differentiation under the integral sign\nIt's possible to prove the result using elementary calculus by applying the [differentiation under the integral sign](https://bliptext.com/articles/leibniz-integral-rule) technique to an integral due to Freitas:\nWhile the [primitive function](https://bliptext.com/articles/primitive-function) of the integrand cannot be expressed in terms of elementary functions, by differentiating with respect to \\alpha we arrive at\nwhich can be integrated by [substituting](https://bliptext.com/articles/integration-by-substitution) u=e^{-x} and decomposing into [partial fractions](https://bliptext.com/articles/partial-fractions).\nIn the range the definite integral reduces to\nThe expression can be simplified using the [arctangent addition formula](https://bliptext.com/articles/inverse-trigonometric-functions) and integrated with respect to \\alpha by means of [trigonometric substitution](https://bliptext.com/articles/trigonometric-substitution), resulting in\nThe [integration constant](https://bliptext.com/articles/integration-constant) c can be determined by noticing that two distinct values of I(\\alpha) are related by\nbecause when calculating I(2) we can [factor](https://bliptext.com/articles/factorization) and express it in terms of I(0) using the [logarithm of a power identity](https://bliptext.com/articles/list-of-logarithmic-identities) and the [substitution](https://bliptext.com/articles/integration-by-substitution) u=x/2.\nThis makes it possible to determine, and it follows that\nThis final integral can be evaluated by expanding the natural logarithm into its [Taylor series](https://bliptext.com/articles/mercator-series):\nThe last two identities imply\n\n# Cauchy's proof\nWhile most proofs use results from advanced [mathematics](https://bliptext.com/articles/mathematics), such as [Fourier analysis](https://bliptext.com/articles/fourier-analysis), [complex analysis](https://bliptext.com/articles/complex-analysis), and [multivariable [calculus](https://bliptext.com/articles/calculus)](https://bliptext.com/articles/multivariable-[calculus](https://bliptext.com/articles/calculus)), the following does not even require single-variable [calculus](https://bliptext.com/articles/calculus) (until a single [limit](https://bliptext.com/articles/limit-of-a-function) is taken at the end).\nFor a proof using the residue theorem, see [here](https://bliptext.com/articles/residue-theorem).\n\n## History of this proof\nThe proof goes back to [Augustin Louis Cauchy](https://bliptext.com/articles/augustin-louis-cauchy) (Cours d'Analyse, 1821, Note VIII).\nIn 1954, this proof appeared in the book of [Akiva](https://bliptext.com/articles/akiva-yaglom) and [Isaak Yaglom](https://bliptext.com/articles/isaak-yaglom) \"Nonelementary Problems in an Elementary Exposition\".\nLater, in 1982, it appeared in the journal Eureka, attributed to John Scholes, but Scholes claims he learned the proof from [Peter Swinnerton-Dyer](https://bliptext.com/articles/peter-swinnerton-dyer), and in any case he maintains the proof was \"common knowledge at [Cambridge](https://bliptext.com/articles/university-of-cambridge) in the late 1960s\".\n\n## The proof\n[[File:limit circle FbN.jpeg|thumb|The inequality is shown pictorially for any . The three terms are the areas of the triangle OAC, circle section OAB, and the triangle OAB.\nTaking reciprocals and squaring gives\n.]] The main idea behind the proof is to bound the partial (finite) sums\nbetween two expressions, each of which will tend to π2⁄6 as\nm\napproaches infinity.\nThe two expressions are derived from identities involving the [cotangent](https://bliptext.com/articles/cotangent) and [cosecant](https://bliptext.com/articles/cosecant) functions.\nThese identities are in turn derived from [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula), and we now turn to establishing these identities.\nLet\nx\nbe a real number with\n0 < x < π⁄2\n, and let\nn\nbe a positive odd integer.\nThen from de Moivre's formula and the definition of the cotangent function, we have\nFrom the [binomial theorem](https://bliptext.com/articles/binomial-theorem), we have\nCombining the two equations and equating imaginary parts gives the identity\nWe take this identity, fix a positive integer\nm\n, set\n, and consider\nfor\n. Then\nnxr\nis a multiple of π and therefore\n. So,\nfor every\n. The values\nare distinct numbers in the interval\n0 <\nxr\n< π⁄2\n. Since the function\ncot2 x\nis [one-to-one](https://bliptext.com/articles/injective-function) on this interval, the numbers\nare distinct for\n. By the above equation, these\nm\nnumbers are the roots of the\nm\nth degree polynomial\nBy [Vieta's formulas](https://bliptext.com/articles/vieta-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\nSubstituting the [identity](https://bliptext.com/articles/list-of-trigonometric-identities)\n, we have\nNow consider the inequality\ncot2 x < 1⁄x2 < csc2 x\n(illustrated geometrically above).\nIf we add up all these inequalities for each of the numbers\n, and if we use the two identities above, we get\nMultiplying through by\n( π⁄2m + 1 ) 2\n, this becomes\nAs\nm\napproaches infinity, the left and right hand expressions each approach π2⁄6, so by the [squeeze theorem](https://bliptext.com/articles/squeeze-theorem),\nand this completes the proof.\n\n# Proof assuming Weil's conjecture on Tamagawa numbers\nA proof is also possible assuming [Weil's conjecture on Tamagawa numbers](https://bliptext.com/articles/weil-s-conjecture-on-tamagawa-numbers).\nThe conjecture asserts for the case of the [algebraic group](https://bliptext.com/articles/algebraic-group) SL2(**R**) that the [Tamagawa number](https://bliptext.com/articles/tamagawa-number) of the group is one.\nThat is, the quotient of the special linear group over the rational [adeles](https://bliptext.com/articles/adele-ring) by the special linear group of the rationals (a [compact set](https://bliptext.com/articles/compact-set), because is a lattice in the [adeles](https://bliptext.com/articles/adele-ring)) has Tamagawa measure 1:\nTo determine a Tamagawa measure, the group SL_2 consists of matrices\nwith xt-yz=1.\nAn invariant [volume form](https://bliptext.com/articles/volume-form) on the group is\nThe measure of the quotient is the product of the measures of corresponding to the infinite place, and the measures of in each finite place, where \\mathbb Z_p is the [p-adic integers](https://bliptext.com/articles/p-adic-integers).\nFor the local factors,\nwhere F_p is the field with p elements, and is the [congruence subgroup](https://bliptext.com/articles/congruence-subgroup) modulo p. Since each of the coordinates x,y,z map the latter group onto and, the measure of is , where \\mu_p is the normalized [Haar measure](https://bliptext.com/articles/haar-measure) on \\mathbb Z_p.\nAlso, a standard computation shows that.\nPutting these together gives.\nAt the infinite place, an integral computation over the fundamental domain of shows that, and therefore the Weil conjecture finally gives\nOn the right-hand side, we recognize the [Euler product](https://bliptext.com/articles/euler-product) for 1/\\zeta(2), and so this gives the solution to the Basel problem.\nThis approach shows the connection between (hyperbolic) geometry and arithmetic, and can be inverted to give a proof of the Weil conjecture for the special case of SL_2, contingent on an independent proof that.\n\n# Geometric proof\nThe Basel problem can be proved with [Euclidean geometry](https://bliptext.com/articles/euclidean-geometry), using the insight that the real line can be seen as a circle of infinite radius.\nAn intuitive, if not completely rigorous sketch is given here.\nthat is,\nor\n.\nThe independence of the sisc from N can be proved easily with Euclidean geometry for the more restrictive case where N is a power of 2, i.e. N = 2^n, which still allows the limiting argument to be applied.\nThe proof proceeds by [induction](https://bliptext.com/articles/mathematical-induction) on n, and uses the [Inverse Pythagorean Theorem](https://bliptext.com/articles/inverse-pythagorean-theorem), which states that:\nwhere a and b are the cathetes and h is the height of a right triangle.\n# [The sum of inverse squares of distances of P1 and P2 from Q equals the inverse square distance from P to Q. | upload.wikimedia.org/wikipedia/commons/3/3e/Induction///step///1728668638527.jpg]\n\n# Other identities\nSee the special cases of the identities for the [Riemann zeta function](https://bliptext.com/articles/riemann-zeta-function) when s = 2.\nOther notably special identities and representations of this constant appear in the sections below.\n\n## Series representations\nThe following are series representations of the constant:\nThere are also [BBP-type](https://bliptext.com/articles/bailey-borwein-plouffe-formula) series expansions for\nζ(2)\n.\n\n## Integral representations\nThe following are integral representations of\n\n## Continued fractions\nIn van der Poorten's classic article chronicling [Apéry's proof of the irrationality of \\zeta(3)](https://bliptext.com/articles/ap-ry-s-constant), the author notes as \"a red herring\" the similarity of a [simple continued fraction](https://bliptext.com/articles/simple-continued-fraction) for Apery's constant, and the following one for the Basel constant:\nwhere.\nAnother continued fraction of a similar form is:\nwhere.","revisionCount":0},"hasMore":false,"nextCursor":null}