The sum of the series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct. He produced an accepted proof in 1741.
The solution to this problem can be used to estimate the probability that two large random numbers are coprime. Two random integers in the range from 1 to , in the limit as goes to infinity, are relatively prime with a probability that approaches , the reciprocal of the solution to the Basel problem.
Euler's original derivation of the value 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.
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 . 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 
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 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.
Generalizations of Euler's method using elementary symmetric polynomialsEdit
The Riemann zeta functionζ(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 numbers 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:
A proof using Euler's formula and L'Hôpital's ruleEdit
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 inequality is shown pictorially for any . The three terms are the areas of the triangle OAC, circle section OAB, and the triangle OAB. Taking reciprocals and squaring gives .
The main idea behind the proof is to bound the partial (finite) sums
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
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 xr = rπ/2m + 1 for r = 1, 2, ..., m. Then nxr is a multiple of π and therefore sin(nxr) = 0. So,
for every r = 1, 2, ..., m. The values xr = x1, x2, ..., xm are distinct numbers in the interval 0 < xr < π/2. Since the function cot2x is one-to-one on this interval, the numbers tr = cot2xr 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 identitycsc2x = cot2x + 1, we have
Now consider the inequality cot2x < 1/x2 < csc2x (illustrated geometrically above). If we add up all these inequalities for each of the numbers xr = 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.
Proof assuming Weil's conjecture on Tamagawa numbersEdit
The 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 is the p-adic integers.
For the local factors,
where is the field with elements, and is the congruence subgroup modulo . Since each of the coordinates map the latter group onto and , the measure of is , where is the normalized Haar measure on . Also, a standard computation shows that . Putting these together gives .
At the infinite place, an integral computation over the fundamental domain of shows that , and therefore the Weil conjecture finally gives
On the right-hand side, we recognize the Euler product for , 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 , contingent on an independent proof that .
See the special cases of the identities for the Riemann zeta function when Other notably special identities and representations of this constant appear in the sections below.
The following are series representations of the constant:
In van der Poorten's classic article chronicling Apéry's proof of the irrationality of , the author notes several parallels in proving the irrationality of 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
^Havil, J. (2003). Gamma: Exploring Euler's Constant. Princeton, New Jersey: Princeton University Press. pp. 37–42 (Chapter 4). ISBN 0-691-09983-9.
^Cf., the formulae for generalized Stirling numbers proved in: Schmidt, M. D. (2018). "Combinatorial Identities for Generalized Stirling Numbers Expanding f-Factorial Functions and the f-Harmonic Numbers". J. Integer Seq. 21 (Article 18.2.7).
^Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. p. 61. ISBN 978-4-431-54919-2.
^Ransford, T J (Summer 1982). "An Elementary Proof of " (PDF). Eureka. 42 (1): 3–4.
^Vladimir Platonov; Andrei Rapinchuk (1994), Algebraic groups and number theory, translated by Rachel Rowen, Academic Press|
^ abWeisstein, Eric W. "Riemann Zeta Function \zeta(2)". MathWorld. Retrieved 29 April 2018.
^Connon, D. F. (2007). "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers (Volume I)". arXiv:0710.4022 [math.HO].
^Weisstein, Eric W. "Double Integral". MathWorld. Retrieved 29 April 2018.
^Weisstein, Eric W. "Hadjicostas's Formula". MathWorld. Retrieved 29 April 2018.