Euler's original derivation of the value ${\displaystyle \pi ^{2}/6}$  essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.

Of course, Euler's original reasoning requires justification (100 years later, Karl Weierstrass 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 expansion of the sine function

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 for expanding an infinite degree polynomial in terms of its roots, but in fact is not always true for general ${\displaystyle P(x)}$ .^[5] This factorization expands the equation into:

If we formally multiply out this product and collect all the x² terms (we are allowed to do so because of Newton's identities), we see by induction that the x² coefficient of sin x/x is ^[6]

But from the original infinite series expansion of sin x/x, the coefficient of x² is −1/3! = −1/6. These two coefficients must be equal; thus,

Multiplying both sides of this equation by −π² gives the sum of the reciprocals of the positive square integers.

This method of calculating ${\displaystyle \zeta (2)}$  is detailed in expository fashion most notably in Havil's Gamma book which details many zeta function and logarithm-related series and integrals, as well as a historical perspective, related to the Euler gamma constant.^[7]

Generalizations of Euler's method using elementary symmetric polynomialsEdit

Using formulae obtained from elementary symmetric polynomials,^[8] 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:

For example, let the partial product for ${\displaystyle \sin(x)}$  expanded as above be defined by ${\displaystyle {\frac {S_{n}(x)}{x}}:=\prod \limits _{k=1}^{n}\left(1-{\frac {x^{2}}{k^{2}\cdot \pi ^{2}}}\right)}$ . 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

and so on for subsequent coefficients of ${\displaystyle [x^{2k}]{\frac {S_{n}(x)}{x}}}$ . There are other forms of Newton's identities expressing the (finite) power sums ${\displaystyle H_{n}^{(2k)}}$  in terms of the elementary symmetric polynomials, ${\displaystyle e_{i}\equiv e_{i}\left(-{\frac {\pi ^{2}}{1^{2}}},-{\frac {\pi ^{2}}{2^{2}}},-{\frac {\pi ^{2}}{3^{2}}},-{\frac {\pi ^{2}}{4^{2}}},\ldots \right),}$  but we can go a more direct route to expressing non-recursive formulas for ${\displaystyle \zeta (2k)}$  using the method of elementary symmetric polynomials. Namely, we have a recurrence relation between the elementary symmetric polynomials and the power sum polynomials given as on this page by

which in our situation equates to the limiting recurrence relation (or generating function convolution, or product) expanded as

Then by differentiation and rearrangement of the terms in the previous equation, we obtain that

Consequences of Euler's proofEdit

By the above results, we can conclude that ${\displaystyle \zeta (2k)}$  is always a rational multiple of ${\displaystyle \pi ^{2k}}$ . In particular, since ${\displaystyle \pi }$  and integer powers of it are transcendental, we can conclude at this point that ${\displaystyle \zeta (2k)}$  is irrational, and more precisely, transcendental for all ${\displaystyle k\geq 1}$ . By contrast, the properties of the odd-indexed zeta constants, including Apéry's constant ${\displaystyle \zeta (3)}$ , are almost completely unknown.

The Riemann zeta function ζ(s) is one of the most significant functions in mathematics because of its relationship to the distribution of the prime numbers. The zeta function is defined for any complex number s with real part greater than 1 by the following formula:

Taking s = 2, we see that ζ(2) is equal to the sum of the reciprocals of the squares of all positive integers:

Convergence can be proven by the integral test, or by the following inequality:

This gives us the upper bound 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 ζ(s) has a simple expression in terms of the Bernoulli numbers whenever s is a positive even integer. With s = 2n:^[9]

The normalized sinc function ${\displaystyle {\text{sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}}$  has a Weierstrass factorization representation as an infinite product:

The infinite product is analytic, so taking the natural logarithm of both sides and differentiating yields

(by uniform convergence, the interchange of the derivative and infinite series is permissible). After dividing the equation by ${\displaystyle 2x}$  and regrouping one gets

We make a change of variables (${\displaystyle x=-it}$ ):

Euler's formula can be used to deduce that

Then

Now we take the limit as ${\displaystyle t}$  approaches zero and use L'Hôpital's rule thrice. By Tannery's theorem applied to ${\textstyle \lim _{t\to \infty }\sum _{n=1}^{\infty }1/(n^{2}+1/t^{2})}$ , we can interchange the limit and infinite series so that ${\textstyle \lim _{t\to 0}\sum _{n=1}^{\infty }1/(n^{2}+t^{2})=\sum _{n=1}^{\infty }1/n^{2}}$  and by L'Hôpital's rule

Use Parseval's identity (applied to the function f(x) = x) to obtain

for n ≠ 0, and c₀ = 0. Thus,

and

Therefore,

Given a complete orthonormal basis in the space ${\displaystyle L_{\operatorname {per} }^{2}(0,1)}$  of L2 periodic functions over ${\displaystyle (0,1)}$  (i.e., the subspace of square-integrable functions which are also periodic), denoted by ${\displaystyle \{e_{i}\}_{i=-\infty }^{\infty }}$ , Parseval's identity tells us that

where ${\displaystyle \|x\|:={\sqrt {\langle x,x\rangle }}}$  is defined in terms of the inner product on this Hilbert space given by

We can consider the orthonormal basis on this space defined by ${\displaystyle e_{k}\equiv e_{k}(\vartheta ):=\exp(2\pi \imath k\vartheta )}$  such that ${\displaystyle \langle e_{k},e_{j}\rangle =\int _{0}^{1}e^{2\pi \imath (k-j)\vartheta }\,d\vartheta =\delta _{k,j}}$ . Then if we take ${\displaystyle f(\vartheta ):=\vartheta }$ , we can compute both that

by elementary calculus and integration by parts, respectively. Finally, by Parseval's identity stated in the form above, we obtain that

Generalizations and recurrence relationsEdit

Note that by considering higher-order powers of ${\displaystyle f_{j}(\vartheta ):=\vartheta ^{j}\in L_{\operatorname {per} }^{2}(0,1)}$  we can use integration by parts to extend this method to enumerating formulas for ${\displaystyle \zeta (2j)}$  when ${\displaystyle j>1}$ . In particular, suppose we let

so that integration by parts yields the recurrence relation that

Then by applying Parseval's identity as we did for the first case above along with the linearity of the inner product yields that

While most proofs use results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (until a single limit is taken at the end).

For a proof using the residue theorem, see here.

History of this proofEdit

The proof goes back to Augustin Louis Cauchy (Cours d'Analyse, 1821, Note VIII). In 1954, this proof appeared in the book of Akiva and Isaak Yaglom "Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal Eureka,^[10] attributed to John Scholes, but Scholes claims he learned the proof from Peter Swinnerton-Dyer, and in any case he maintains the proof was "common knowledge at Cambridge in the late 1960s".^{[citation needed]}

The proofEdit

The main idea behind the proof is to bound the partial (finite) sums

Let x be a real number with 0 < x < π/2, and let n be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we have

From the binomial theorem, we have

Combining the two equations and equating imaginary parts gives the identity

We take this identity, fix a positive integer m, set n = 2m + 1, and consider x_r = rπ/2m + 1 for r = 1, 2, ..., m. Then nx_r is a multiple of π and therefore sin(nx_r) = 0. So,

for every r = 1, 2, ..., m. The values x_r = x₁, x₂, ..., x_m are distinct numbers in the interval 0 < x_r < π/2. Since the function cot² x is one-to-one on this interval, the numbers t_r = cot² x_r are distinct for r = 1, 2, ..., m. By the above equation, these m numbers are the roots of the mth degree polynomial

By 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

Substituting the identity csc² x = cot² x + 1, we have

Now consider the inequality cot² x < 1/x² < csc² x (illustrated geometrically above). If we add up all these inequalities for each of the numbers x_r = rπ/2m + 1, and if we use the two identities above, we get

Multiplying through by (π/2m + 1)²
, this becomes

As m approaches infinity, the left and right hand expressions each approach π²/6, so by the squeeze theorem,

and this completes the proof.

A proof is also possible assuming Weil's conjecture on Tamagawa numbers.^[11] The conjecture asserts for the case of the algebraic group SL₂(R) that the Tamagawa number of the group is one. That is, the quotient of the special linear group over the rational adeles by the special linear group of the rationals (a compact set, because ${\displaystyle SL_{2}(\mathbb {Q} )}$  is a lattice in the adeles) has Tamagawa measure 1:

To determine a Tamagawa measure, the group ${\displaystyle SL_{2}}$  consists of matrices

The measure of the quotient is the product of the measures of ${\displaystyle SL_{2}(\mathbb {Z} )\setminus SL_{2}(\mathbb {R} )}$  corresponding to the infinite place, and the measures of ${\displaystyle SL_{2}(\mathbb {Z} _{p})}$  in each finite place, where ${\displaystyle \mathbb {Z} _{p}}$  is the p-adic integers.

For the local factors,

At the infinite place, an integral computation over the fundamental domain of ${\displaystyle SL_{2}(\mathbb {Z} )}$  shows that ${\displaystyle \omega (SL_{2}(\mathbb {Z} )\setminus SL_{2}(\mathbb {R} )=\pi ^{2}/6}$ , and therefore the Weil conjecture finally gives

This 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 ${\displaystyle SL_{2}}$ , contingent on an independent proof that ${\displaystyle \zeta (2)=\pi ^{2}/6}$ .

See the special cases of the identities for the Riemann zeta function when ${\displaystyle s=2.}$  Other notably special identities and representations of this constant appear in the sections below.

Series representationsEdit

The following are series representations of the constant:^[12]

There are also BBP-type series expansions for ζ(2).^[12]

Integral representationsEdit

The following are integral representations of ${\displaystyle \zeta (2){\text{:}}}$ ^[13][14][15]

Continued fractionsEdit

In van der Poorten's classic article chronicling Apéry's proof of the irrationality of ${\displaystyle \zeta (3)}$ ,^[16] the author notes several parallels in proving the irrationality of ${\displaystyle \zeta (2)}$  to Apéry's proof. In particular, he documents recurrence relations for almost integer sequences converging to the constant and continued fractions for the constant. Other continued fractions for this constant include^[17]

and^{[18][unreliable source?]}

where ${\displaystyle v_{n}=2n-1\mapsto \{1,3,5,7,9,\ldots \}}$  and ${\displaystyle {\widetilde {v}}_{n}=11n^{2}-11n+3\mapsto \{3,25,69,135,\ldots \}}$ .

• Riemann zeta function
• Apéry's constant
• List of sums of reciprocals

• Weil, André (1983), Number Theory: An Approach Through History, Springer-Verlag, ISBN 0-8176-3141-0.
• Dunham, William (1999), Euler: The Master of Us All, Mathematical Association of America, ISBN 0-88385-328-0.
• Derbyshire, John (2003), Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, ISBN 0-309-08549-7.
• Aigner, Martin; Ziegler, Günter M. (1998), Proofs from THE BOOK, Berlin, New York: Springer-Verlag
• Edwards, Harold M. (2001), Riemann's Zeta Function, Dover, ISBN 0-486-41740-9.

• An infinite series of surprises by C. J. Sangwin
• From ζ(2) to Π. The Proof. step-by-step proof
• "Remarques sur un beau rapport entre les series des puissances tant directes que reciproques" (PDF)., English translation with notes of Euler's paper by Lucas Willis and Thomas J. Osler
• Ed Sandifer. "How Euler did it" (PDF).
• James A. Sellers (February 5, 2002). "Beyond Mere Convergence" (PDF). Retrieved 2004-02-27.
• Robin Chapman. "Evaluating ζ(2)" (PDF). (fourteen proofs)
• Visualization of Euler's factorization of the sine function
• Johan W Ästlund (December 8, 2010). "Summing inverse squares by Euclidean geometry" (PDF).
  • Why is pi here? And why is it squared? A geometric answer to the Basel problem on YouTube (animated proof based on the above)