Basel problem

The Riemann zeta function

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: $${\displaystyle \zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}.}$$

Taking s = 2, we see that ζ(2) is equal to the sum of the reciprocals of the squares of all positive integers: $${\displaystyle \zeta (2)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\frac{{\pi }^2}{6}\approx \mathrm{1.644934.}}$$

Convergence can be proven by the integral test, or by the following inequality: $${\displaystyle \begin{array}{cc}\sum _{n=1}^N\frac{1}{n^2}& <1+\sum _{n=2}^N\frac{1}{n(n-1)}\\ & =1+\sum _{n=2}^N\left(\frac{1}{n-1}-\frac{1}{n}\right)\\ & =1+1-\frac{1}{N}\stackrel{N\to \mathrm{\infty }}{⟶}2.\end{array}}$$

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: $${\displaystyle \zeta (2n)=\frac{(2\pi {)}^{2n}(-1{)}^{n+1}{B}_{2n}}{2\cdot (2n)!}.}$$

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

The normalized sinc function ${\displaystyle \text{sinc}(x)=\frac{\sin (\pi x)}{\pi x}}$ has a Weierstrass factorization representation as an infinite product: $${\displaystyle \frac{\sin (\pi x)}{\pi x}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{x^2}{n^2}\right).}$$

The infinite product is analytic, so taking the natural logarithm of both sides and differentiating yields $${\displaystyle \frac{\pi \cos (\pi x)}{\sin (\pi x)}-\frac{1}{x}=-\sum _{n=1}^{\mathrm{\infty }}\frac{2x}{n^2-x^2}}$$

(by uniform convergence, the interchange of the derivative and infinite series is permissible). After dividing the equation by ${\displaystyle 2x}$ and regrouping one gets $${\displaystyle \frac{1}{2x^2}-\frac{\pi \cot (\pi x)}{2x}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2-x^2}.}$$

We make a change of variables (${\displaystyle x=-it}$): $${\displaystyle -\frac{1}{2t^2}+\frac{\pi \cot (-\pi it)}{2it}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+t^2}.}$$

Euler's formula can be used to deduce that $${\displaystyle \frac{\pi \cot (-\pi it)}{2it}=\frac{\pi }{2it}\frac{i\left(e^{2\pi t}+1\right)}{e^{2\pi t}-1}=\frac{\pi }{2t}+\frac{\pi }{t\left(e^{2\pi t}-1\right)}.}$$ or using the corresponding hyperbolic function: $${\displaystyle \frac{\pi \cot (-\pi it)}{2it}=\frac{\pi }{2t}i\cot (\pi it)=\frac{\pi }{2t}\coth (\pi t).}$$

Then $${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+t^2}=\frac{\pi \left(t{e}^{2\pi t}+t\right)-e^{2\pi t}+1}{2\left(t^2{e}^{2\pi t}-t^2\right)}=-\frac{1}{2t^2}+\frac{\pi }{2t}\coth (\pi t).}$$

Now we take the limit as ${\displaystyle t}$ approaches zero and use L'Hôpital's rule thrice. By Tannery's theorem applied to $\underset{t\to \mathrm{\infty }}{lim}\sum _{n=1}^{\mathrm{\infty }}1/(n^2+1/t^2)$, we can interchange the limit and infinite series so that $\underset{t\to 0}{lim}\sum _{n=1}^{\mathrm{\infty }}1/(n^2+t^2)=\sum _{n=1}^{\mathrm{\infty }}1/n^2$ and by L'Hôpital's rule $${\displaystyle \begin{array}{cc}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}& =\underset{t\to 0}{lim}\frac{\pi }{4}\frac{2\pi t{e}^{2\pi t}-e^{2\pi t}+1}{\pi t^2{e}^{2\pi t}+t{e}^{2\pi t}-t}\\ & =\underset{t\to 0}{lim}\frac{{\pi }^{3}t{e}^{2\pi t}}{2\pi \left(\pi t^2{e}^{2\pi t}+2t{e}^{2\pi t}\right)+e^{2\pi t}-1}\\ & =\underset{t\to 0}{lim}\frac{{\pi }^2(2\pi t+1)}{4{\pi }^2{t}^2+12\pi t+6}\\ & =\frac{{\pi }^2}{6}.\end{array}}$$

A proof using Fourier series

Use Parseval's identity (applied to the function f(x) = x) to obtain $${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}|c_n{|}^2=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{x}^{2}dx,}$$ where $${\displaystyle \begin{array}{cc}{c}_n& =\frac{1}{2\pi }{\int }_{-\pi }^{\pi }x{e}^{-inx}dx\\ & =\frac{n\pi \cos (n\pi )-\sin (n\pi )}{\pi n^2}i\\ & =\frac{\cos (n\pi )}{n}i\\ & =\frac{(-1{)}^n}{n}i\end{array}}$$

for n ≠ 0, and c₀ = 0. Thus, $${\displaystyle |c_n{|}^2=\{\begin{array}{ll}{\displaystyle \frac{1}{n^2}},& \text{for }n\ne 0,\\ 0,& \text{for }n=0,\end{array}}$$

and $${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}|c_n{|}^2=2\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{x}^{2}dx.}$$

Therefore, $${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{1}{4\pi }{\int }_{-\pi }^{\pi }{x}^{2}dx=\frac{{\pi }^2}{6}}$$ as required.

Another proof using Parseval's identity

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

where ${\displaystyle \Vert x\Vert :=\sqrt{⟨x,x⟩}}$ is defined in terms of the inner product on this Hilbert space given by $${\displaystyle ⟨f,g⟩={\int }_0^{1}f(x)\overline{g(x)}dx,f,g\in L_{\mathrm{per}}^2(0,1).}$$

We can consider the orthonormal basis on this space defined by ${\displaystyle e_k\equiv e_k(\vartheta ):=\exp (2\pi ık\vartheta )}$ such that ${\displaystyle ⟨e_k,e_j⟩={\int }_0^1{e}^{2\pi ı(k-j)\vartheta }d\vartheta ={\delta }_{k,j}}$. Then if we take ${\displaystyle f(\vartheta ):=\vartheta }$, we can compute both that $${\displaystyle \begin{array}{cc}\Vert f{\Vert }^2& ={\int }_0^1{\vartheta }^{2}d\vartheta =\frac{1}{3}\\ ⟨f,e_k⟩& ={\int }_0^1\vartheta e^{-2\pi ık\vartheta }d\vartheta =\{\begin{array}{ll}\frac{1}{2},& k=0\\ -\frac{1}{2\pi ık}& k\ne 0,\end{array}\end{array}}$$

by elementary calculus and integration by parts, respectively. Finally, by Parseval's identity stated in the form above, we obtain that $${\displaystyle \begin{array}{cc}\Vert f{\Vert }^2=\frac{1}{3}& =\sum _{\stackrel{k=-\mathrm{\infty }}{k\ne 0}}^{\mathrm{\infty }}\frac{1}{(2\pi k{)}^2}+\frac{1}{4}=2\sum _{k=1}^{\mathrm{\infty }}\frac{1}{(2\pi k{)}^2}+\frac{1}{4}\\ & ⟹\frac{{\pi }^2}{6}=\frac{2{\pi }^2}{3}-\frac{{\pi }^2}{2}=\zeta (2).\end{array}}$$

Generalizations and recurrence relations

Note that by considering higher-order powers of ${\displaystyle f_j(\vartheta ):={\vartheta }^j\in L_{\mathrm{per}}^2(0,1)}$ 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 $${\displaystyle I_{j,k}:={\int }_0^1{\vartheta }^j{e}^{-2\pi ık\vartheta }d\vartheta ,}$$

so that integration by parts yields the recurrence relation that $${\displaystyle \begin{array}{cc}{I}_{j,k}& =\{\begin{array}{ll}\frac{1}{j+1},& k=0;\\ -\frac{1}{2\pi ı\cdot k}+\frac{j}{2\pi ı\cdot k}{I}_{j-1,k},& k\ne 0\end{array}\\ & =\{\begin{array}{ll}\frac{1}{j+1},& k=0;\\ -\sum _{m=1}^j\frac{j!}{(j+1-m)!}\cdot \frac{1}{(2\pi ı\cdot k{)}^m},& k\ne 0.\end{array}\end{array}}$$

Then by applying Parseval's identity as we did for the first case above along with the linearity of the inner product yields that $${\displaystyle \begin{array}{cc}\Vert f_j{\Vert }^2=\frac{1}{2j+1}& =2\sum _{k\ge 1}{I}_{j,k}{\overline{I}}_{j,k}+\frac{1}{(j+1{)}^2}\\ & =2\sum _{m=1}^j\sum _{r=1}^j\frac{j{!}^2}{(j+1-m)!(j+1-r)!}\frac{(-1{)}^r}{{ı}^{m+r}}\frac{\zeta (m+r)}{(2\pi {)}^{m+r}}+\frac{1}{(j+1{)}^2}.\end{array}}$$

Proof using differentiation under the integral sign

It's possible to prove the result using elementary calculus by applying the differentiation under the integral sign technique to an integral due to Freitas: $${\displaystyle I(\alpha )={\int }_0^{\mathrm{\infty }}\ln \left(1+\alpha e^{-x}+e^{-2x}\right)dx.}$$

While the primitive function of the integrand cannot be expressed in terms of elementary functions, by differentiating with respect to ${\displaystyle \alpha }$ we arrive at

$${\displaystyle \frac{dI}{d\alpha }={\int }_0^{\mathrm{\infty }}\frac{e^{-x}}{1+\alpha e^{-x}+e^{-2x}}dx,}$$ which can be integrated by substituting ${\displaystyle u=e^{-x}}$ and completing the square. In the range ${\displaystyle -2<\alpha <2}$ the definite integral reduces to

$${\displaystyle \frac{dI}{d\alpha }=\frac{2}{\sqrt{4-{\alpha }^2}}\left[\arctan \left(\frac{\alpha +2}{\sqrt{4-{\alpha }^2}}\right)-\arctan \left(\frac{\alpha }{\sqrt{4-{\alpha }^2}}\right)\right].}$$

The expression can be simplified using the arctangent addition formula and integrated with respect to ${\displaystyle \alpha }$ by means of trigonometric substitution, resulting in

$${\displaystyle I(\alpha )=-\frac{1}{2}\arccos {\left(\frac{\alpha }{2}\right)}^2+c.}$$

The integration constant ${\displaystyle c}$ can be determined by noticing that two distinct values of ${\displaystyle I(\alpha )}$ are related by

$${\displaystyle I(2)=4I(0),}$$ because when calculating ${\displaystyle I(2)}$ we can factor ${\displaystyle 1+2e^{-x}+e^{-2x}=(1+e^{-x}{)}^2}$ and express it in terms of ${\displaystyle I(0)}$ using the logarithm of a power identity and the substitution ${\displaystyle u=x/2}$. This makes it possible to determine ${\displaystyle c=\frac{{\pi }^2}{6}}$, and it follows that

$${\displaystyle I(-2)=2{\int }_0^{\mathrm{\infty }}\ln (1-e^{-x})dx=-\frac{{\pi }^2}{3}.}$$

This final integral can be evaluated by expanding the natural logarithm into its Taylor series:

$${\displaystyle {\int }_0^{\mathrm{\infty }}\ln (1-e^{-x})dx=-\sum _{n=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{e^{-nx}}{n}dx=-\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}.}$$

The last two identities imply

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}.}$$

Cauchy's proof

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 proof

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, 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".

The proof

The main idea behind the proof is to bound the partial (finite) sums $${\displaystyle \sum _{k=1}^m\frac{1}{k^2}=\frac{1}{1^2}+\frac{1}{2^2}+\cdots +\frac{1}{m^2}}$$ between two expressions, each of which will tend to ⁠π²/6⁠ as m approaches infinity. The two expressions are derived from identities involving the cotangent and cosecant functions. These identities are in turn derived from de Moivre's formula, and we now turn to establishing these identities.

Let 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 $${\displaystyle \begin{array}{cc}\frac{\cos (nx)+i\sin (nx)}{{\sin }^{n}x}& =\frac{(\cos x+i\sin x{)}^n}{{\sin }^{n}x}\\ & ={\left(\frac{\cos x+i\sin x}{\sin x}\right)}^n\\ & =(\cot x+i{)}^n.\end{array}}$$

From the binomial theorem, we have $${\displaystyle \begin{array}{cc}(\cot x+i{)}^n=& (\genfrac{}{}{0ex}{}{n}{0}){\cot }^{n}x+(\genfrac{}{}{0ex}{}{n}{1})({\cot }^{n-1}x)i+\cdots +(\genfrac{}{}{0ex}{}{n}{n-1})(\cot x)i^{n-1}+(\genfrac{}{}{0ex}{}{n}{n})i^n\\ =& ((\genfrac{}{}{0ex}{}{n}{0}){\cot }^{n}x-(\genfrac{}{}{0ex}{}{n}{2}){\cot }^{n-2}x\pm \cdots )+i((\genfrac{}{}{0ex}{}{n}{1}){\cot }^{n-1}x-(\genfrac{}{}{0ex}{}{n}{3}){\cot }^{n-3}x\pm \cdots ).\end{array}}$$

Combining the two equations and equating imaginary parts gives the identity $${\displaystyle \frac{\sin (nx)}{{\sin }^{n}x}=((\genfrac{}{}{0ex}{}{n}{1}){\cot }^{n-1}x-(\genfrac{}{}{0ex}{}{n}{3}){\cot }^{n-3}x\pm \cdots ).}$$

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, $${\displaystyle 0=(\genfrac{}{}{0ex}{}{2m+1}{1}){\cot }^{2m}{x}_r-(\genfrac{}{}{0ex}{}{2m+1}{3}){\cot }^{2m-2}{x}_r\pm \cdots +(-1{)}^m(\genfrac{}{}{0ex}{}{2m+1}{2m+1})}$$

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 $${\displaystyle p(t)=(\genfrac{}{}{0ex}{}{2m+1}{1})t^m-(\genfrac{}{}{0ex}{}{2m+1}{3})t^{m-1}\pm \cdots +(-1{)}^m(\genfrac{}{}{0ex}{}{2m+1}{2m+1}).}$$

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 $${\displaystyle {\cot }^2{x}_1+{\cot }^2{x}_2+\cdots +{\cot }^2{x}_m=\frac{(\genfrac{}{}{0ex}{}{2m+1}{3})}{(\genfrac{}{}{0ex}{}{2m+1}{1})}=\frac{2m(2m-1)}{6}.}$$

Substituting the identity csc² x = cot² x + 1, we have $${\displaystyle {\csc }^2{x}_1+{\csc }^2{x}_2+\cdots +{\csc }^2{x}_m=\frac{2m(2m-1)}{6}+m=\frac{2m(2m+2)}{6}.}$$

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 $${\displaystyle \frac{2m(2m-1)}{6}<{\left(\frac{2m+1}{\pi }\right)}^2+{\left(\frac{2m+1}{2\pi }\right)}^2+\cdots +{\left(\frac{2m+1}{m\pi }\right)}^2<\frac{2m(2m+2)}{6}.}$$

Multiplying through by (⁠π/2m + 1⁠)²
, this becomes $${\displaystyle \frac{{\pi }^2}{6}\left(\frac{2m}{2m+1}\right)\left(\frac{2m-1}{2m+1}\right)<\frac{1}{1^2}+\frac{1}{2^2}+\cdots +\frac{1}{m^2}<\frac{{\pi }^2}{6}\left(\frac{2m}{2m+1}\right)\left(\frac{2m+2}{2m+1}\right).}$$

As m approaches infinity, the left and right hand expressions each approach ⁠π²/6⁠, so by the squeeze theorem, $${\displaystyle \zeta (2)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2}=\underset{m\to \mathrm{\infty }}{lim}\left(\frac{1}{1^2}+\frac{1}{2^2}+\cdots +\frac{1}{m^2}\right)=\frac{{\pi }^2}{6}}$$

and this completes the proof.

Proof assuming Weil's conjecture on Tamagawa numbers

A proof is also possible assuming Weil's conjecture on Tamagawa numbers. 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 ${\displaystyle S{L}_2(\mathbb{Q})}$ is a lattice in the adeles) has Tamagawa measure 1: $${\displaystyle \tau (S{L}_2(\mathbb{Q})\setminus S{L}_2(A_{\mathbb{Q}}))=1.}$$

To determine a Tamagawa measure, the group ${\displaystyle S{L}_2}$ consists of matrices $${\displaystyle \left[\begin{array}{cc}x& y\\ z& t\end{array}\right]}$$ with ${\displaystyle xt-yz=1}$. An invariant volume form on the group is $${\displaystyle \omega =\frac{1}{x}dx\wedge dy\wedge dz.}$$

The measure of the quotient is the product of the measures of ${\displaystyle S{L}_2(\mathbb{Z})\setminus S{L}_2(\mathbb{R})}$ corresponding to the infinite place, and the measures of ${\displaystyle S{L}_2({\mathbb{Z}}_p)}$ in each finite place, where ${\displaystyle {\mathbb{Z}}_p}$ is the p-adic integers.

For the local factors, $${\displaystyle \omega (S{L}_2({\mathbb{Z}}_p))=|S{L}_2(F_p)|\omega (S{L}_2({\mathbb{Z}}_p,p))}$$ where ${\displaystyle F_p}$ is the field with ${\displaystyle p}$ elements, and ${\displaystyle S{L}_2({\mathbb{Z}}_p,p)}$ is the congruence subgroup modulo ${\displaystyle p}$. Since each of the coordinates ${\displaystyle x,y,z}$ map the latter group onto ${\displaystyle p{\mathbb{Z}}_p}$ and ${\displaystyle {\left|\frac{1}{x}\right|}_p=1}$, the measure of ${\displaystyle S{L}_2({\mathbb{Z}}_p,p)}$ is ${\displaystyle {\mu }_p(p{\mathbb{Z}}_p{)}^3=p^{-3}}$, where ${\displaystyle {\mu }_p}$ is the normalized Haar measure on ${\displaystyle {\mathbb{Z}}_p}$. Also, a standard computation shows that ${\displaystyle |S{L}_2(F_p)|=p(p^2-1)}$. Putting these together gives ${\displaystyle \omega (S{L}_2({\mathbb{Z}}_p))=(1-1/p^2)}$.

At the infinite place, an integral computation over the fundamental domain of ${\displaystyle S{L}_2(\mathbb{Z})}$ shows that ${\displaystyle \omega (S{L}_2(\mathbb{Z})\setminus S{L}_2(\mathbb{R}))={\pi }^2/6}$, and therefore the Weil conjecture finally gives $${\displaystyle 1=\frac{{\pi }^2}{6}\prod _p\left(1-\frac{1}{p^2}\right).}$$ On the right-hand side, we recognize the Euler product for ${\displaystyle 1/\zeta (2)}$, and so this gives the solution to the Basel problem.

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 ${\displaystyle S{L}_2}$, contingent on an independent proof that ${\displaystyle \zeta (2)={\pi }^2/6}$.

Geometric proof

The Basel problem can be proved with Euclidean geometry, using the insight that the real line can be seen as a circle of infinite radius. An intuitive, if not completely rigorous, sketch is given here.

• Choose an integer ${\displaystyle N}$, and take ${\displaystyle N}$ equally spaced points on a circle with circumference equal to ${\displaystyle 2N}$. The radius of the circle is ${\displaystyle N/\pi }$ and the length of each arc between two points is ${\displaystyle 2}$. Call the points ${\displaystyle P_{\mathrm{1..}N}}$.
• Take another generic point ${\displaystyle Q}$ on the circle, which will lie at a fraction ${\displaystyle 0<\alpha <1}$ of the arc between two consecutive points (say ${\displaystyle P_1}$ and ${\displaystyle P_2}$ without loss of generality).
• Draw all the chords joining ${\displaystyle Q}$ with each of the ${\displaystyle P_{\mathrm{1..}N}}$ points. Now (this is the key to the proof), compute the sum of the inverse squares of the lengths of all these chords, call it ${\displaystyle sisc}$.
• The proof relies on the notable fact that (for a fixed ${\displaystyle \alpha }$), the ${\displaystyle sisc}$ does not depend on ${\displaystyle N}$. Note that intuitively, as ${\displaystyle N}$ increases, the number of chords increases, but their length increases too (as the circle gets bigger), so their inverse square decreases.
• In particular, take the case where ${\displaystyle \alpha =1/2}$, meaning that ${\displaystyle Q}$ is the midpoint of the arc between two consecutive ${\displaystyle P}$'s. The ${\displaystyle sisc}$ can then be found trivially from the case ${\displaystyle N=1}$, where there is only one ${\displaystyle P}$, and one ${\displaystyle Q}$ on the opposite side of the circle. Then the chord is the diameter of the circle, of length ${\displaystyle 2/\pi }$. The ${\displaystyle sisc}$ is then ${\displaystyle {\pi }^2/4}$.
• When ${\displaystyle N}$ goes to infinity, the circle approaches the real line. If you set the origin at ${\displaystyle Q}$, the points ${\displaystyle P_{\mathrm{1..}N}}$ are positioned at the odd integer positions (positive and negative), since the arcs have length 1 from ${\displaystyle Q}$ to ${\displaystyle P_1}$, and 2 onward. You hence get this variation of the Basel Problem:

$${\displaystyle \sum _{z=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{(2z-1{)}^2}=\frac{{\pi }^2}{4}}$$

• From here, you can recover the original formulation with a bit of algebra, as:

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(2n-1{)}^2}+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(2n{)}^2}=\frac{1}{2}\sum _{z=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{(2z-1{)}^2}+\frac{1}{4}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}}$$

that is,

$${\displaystyle \frac{3}{4}\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{8}}$$

or

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}=\frac{{\pi }^2}{6}}$$.

The independence of the ${\displaystyle sisc}$ from ${\displaystyle N}$ can be proved easily with Euclidean geometry for the more restrictive case where ${\displaystyle N}$ is a power of 2, i.e. ${\displaystyle N=2^n}$, which still allows the limiting argument to be applied. The proof proceeds by induction on ${\displaystyle n}$, and uses the Inverse Pythagorean Theorem, which states that:

$${\displaystyle \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{h^2}}$$

where ${\displaystyle a}$ and ${\displaystyle b}$ are the legs and ${\displaystyle h}$ is the height of a right triangle.

• In the base case of ${\displaystyle n=0}$, there is only 1 chord. In the case of ${\displaystyle \alpha =1/2}$, it corresponds to the diameter and the ${\displaystyle sisc}$ is ${\displaystyle {\pi }^2/4}$ as stated above.
• Now, assume that you have ${\displaystyle 2^n}$ points on a circle with radius ${\displaystyle 2^n/\pi }$ and center ${\displaystyle O}$, and ${\displaystyle 2^{n+1}}$ points on a circle with radius ${\displaystyle 2^{n+1}/\pi }$ and center ${\displaystyle R}$. The induction step consists in showing that these 2 circles have the same ${\displaystyle sisc}$ for a given ${\displaystyle \alpha }$.

• Start by drawing the circles so that they share point ${\displaystyle Q}$. Note that ${\displaystyle R}$ lies on the smaller circle. Then, note that ${\displaystyle 2^{n+1}}$ is always even, and a simple geometric argument shows that you can pick pairs of opposite points ${\displaystyle P_1}$ and ${\displaystyle P_2}$ on the larger circle by joining each pair with a diameter. Furthermore, for each pair, one of the points will be in the "lower" half of the circle (closer to ${\displaystyle Q}$) and the other in the "upper" half.

• The diameter of the bigger circle ${\displaystyle P_1{P}_2}$ cuts the smaller circle at ${\displaystyle R}$ and at another point ${\displaystyle P}$. You can then make the following considerations:
  • ${\displaystyle P_1\hat{Q}{P}_2}$ is a right angle, since ${\displaystyle P_1{P}_2}$ is a diameter.
  • ${\displaystyle Q\hat{P}R}$ is a right angle, since ${\displaystyle QR}$ is a diameter.
  • ${\displaystyle Q\hat{R}{P}_2=Q\hat{R}P}$ is half of ${\displaystyle Q\hat{O}P}$ for the Inscribed Angle Theorem.
  • Hence, the arc ${\displaystyle QP}$ is equal to the arc ${\displaystyle Q{P}_2}$, again because the radius is half.
  • The chord ${\displaystyle QP}$ is the height of the right triangle ${\displaystyle Q{P}_1{P}_2}$, hence for the Inverse Pythagorean Theorem:

$${\displaystyle \frac{1}{{\overline{QP}}^2}=\frac{1}{{\overline{Q{P}_1}}^2}+\frac{1}{{\overline{Q{P}_2}}^2}}$$

• Hence for half of the points on the bigger circle (the ones in the lower half) there is a corresponding point on the smaller circle with the same arc distance from ${\displaystyle Q}$ (since the circumference of the smaller circle is half that of the bigger circle, the last two points closer to ${\displaystyle R}$ must have arc distance 2 as well). Vice versa, for each of the ${\displaystyle 2^n}$ points on the smaller circle, we can build a pair of points on the bigger circle, and all of these points are equidistant and have the same arc distance from ${\displaystyle Q}$.
• Furthermore, the total ${\displaystyle sisc}$ for the bigger circle is the same as the ${\displaystyle sisc}$ for the smaller circle, since each pair of points on the bigger circle has the same inverse square sum as the corresponding point on the smaller circle.

Other identities

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 representations

The following are series representations of the constant: $${\displaystyle \begin{array}{cc}\zeta (2)& =3\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2(\genfrac{}{}{0ex}{}{2k}{k})}\\ & =\sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}\frac{(i-1)!(j-1)!}{(i+j)!}.\end{array}}$$

There are also BBP-type series expansions for ζ(2).

Integral representations

The following are integral representations of ${\displaystyle \zeta (2)\text{:}}$ $${\displaystyle \begin{array}{cc}\zeta (2)& =-{\int }_0^1\frac{\log x}{1-x}dx\\ & ={\int }_0^{\mathrm{\infty }}\frac{x}{e^x-1}dx\\ & ={\int }_0^1\frac{(\log x{)}^2}{(1+x{)}^2}dx\\ & =2+2{\int }_1^{\mathrm{\infty }}\frac{\lfloor x\rfloor -x}{x^3}dx\\ & =\exp \left(2{\int }_2^{\mathrm{\infty }}\frac{\pi (x)}{x(x^2-1)}dx\right)\\ & ={\int }_0^1{\int }_0^1\frac{dxdy}{1-xy}\\ & =\frac{4}{3}{\int }_0^1{\int }_0^1\frac{dxdy}{1-(xy{)}^2}\\ & ={\int }_0^1{\int }_0^1\frac{1-x}{1-xy}dxdy+\frac{2}{3}.\end{array}}$$

Continued fractions

In van der Poorten's classic article chronicling Apéry's proof of the irrationality of ${\displaystyle \zeta (3)}$, the author notes as "a red herring" the similarity of a simple continued fraction for Apery's constant, and the following one for the Basel constant: $${\displaystyle \frac{\zeta (2)}{5}=\frac{1}{{\tilde{v}}_1+\frac{1^4}{{\tilde{v}}_2+\frac{2^4}{{\tilde{v}}_3+\frac{3^4}{{\tilde{v}}_4+\ddots }}}},}$$ where ${\displaystyle {\tilde{v}}_n=11n^2-11n+3\mapsto \{3,25,69,135,\dots \}}$. Another continued fraction of a similar form is: $${\displaystyle \frac{\zeta (2)}{2}=\frac{1}{v_1+\frac{1^4}{v_2+\frac{2^4}{v_3+\frac{3^4}{v_4+\ddots }}}},}$$ where ${\displaystyle v_n=2n-1\mapsto \{1,3,5,7,9,\dots \}}$.