Basel problem

Solved, 1735Analysis

Statement

What is the exact value of the sum of the reciprocals of the perfect squares, $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots$ ? The series was known to converge, but its exact closed form eluded mathematicians for nearly a century.

Euler first announced the result in 1735 using a bold analogy between the infinite product for $\sin x$ and the factorisation of a polynomial by its roots — a step he could not yet justify rigorously. A fully rigorous version came only decades later, once Weierstrass's factorisation theorem (1876) justified representing entire functions as infinite products.

Solved by, 1735Leonhard Euler

Euler's method generalises: he found the closed form $\zeta(2n) = \frac{(-1)^{n+1} B_{2n} (2\pi)^{2n}}{2(2n)!}$ for every positive integer $n$ , where $B_{2n}$ are the Bernoulli numbers (for example $\zeta(4) = \pi^4/90$ ). No comparable closed form is known for the odd values $\zeta(3), \zeta(5), \ldots$ ; Roger Apéry proved in 1978 that $\zeta(3)$ is irrational, but its exact value in terms of $\pi$ remains unknown.

References

Frits Beukers, Eugenio Calabi, Jan A. C. Kolk (1993). Sums of generalized harmonic series and volumes · DOI:10.1007/s12045-015-0241-0
William Dunham (1999). Euler: The Master of Us All
Martin Aigner, Günter M. Ziegler (2018). Proofs from THE BOOK

History of the problem

1650Pietro Mengoli poses the problem of summing the reciprocals of the perfect squares in his Novae Quadraturae Arithmeticae, alongside sums of the reciprocals of triangular numbers (which he does solve).
1689Jakob Bernoulli, in Tractatus de Seriebus Infinitis, proves the series converges (by bounding it with a telescoping series) but cannot find its exact value; he asks the mathematical community for help, and the challenge becomes known by the name of his home city, Basel.
1735Leonhard Euler announces $\sum_{n=1}^{\infty} 1/n^2 = \pi^2/6$ , obtained by comparing the Taylor series of $\sin x / x$ with its factorisation over its zeros $\pm n\pi$ .Leonhard Euler
1741Euler publishes a fully worked derivation and extends the method to evaluate $\zeta(2n)$ in closed form for every positive integer $n$ .Leonhard Euler
1993Frits Beukers, Eugenio Calabi and Jan Kolk give an elementary proof using only a change of variables in a double integral over the unit square, avoiding the infinite-product argument entirely.