Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in "The Saint Petersburg Academy of Sciences". Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.
The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:
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 a truly rigorous proof in 1741.
Euler's approach.
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.
To follow Euler's argument, recall the Taylor series expansion of the sine function
Dividing through by , we have
Using the Weierstrass factorization theorem, it can also be shown that the right-hand side is the product of linear factors given by its roots, just as we do for finite polynomials (which Euler assumed as a heuristic for expanding an infinite degree polynomial in terms of its roots, but in fact is not always true for general formula_4):
If we formally multiply out this product and collect all the terms (we are allowed to do so because of Newton's identities), we see by induction that the coefficient of is 
But from the original infinite series expansion of , the coefficient of is . These two coefficients must be equal; thus,
Multiplying both sides of this equation by −2 gives the sum of the reciprocals of the positive square integers.
This method of calculating formula_9 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 polynomials.
Using formulae obtained from elementary symmetric polynomials, 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 formula_11 expanded as above be defined by formula_12. 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 formula_14. There are other forms of Newton's identities expressing the (finite) power sums formula_15 in terms of the elementary symmetric polynomials, formula_16 but we can go a more direct route to expressing non-recursive formulas for formula_17 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 proof.
By Euler's proof for formula_9 explained above and the extension of his method by elementary symmetric polynomials in the previous subsection, we can conclude that formula_17 is "always" a rational multiple of formula_23. Thus compared to the relatively unknown, or at least unexplored up to this point, properties of the odd-indexed zeta constants, including Apéry's constant formula_24, we can conclude much more about this class of zeta constants. In particular, since formula_25 and integer powers of it are transcendental, we can conclude at this point that formula_17 is irrational, and more precisely, transcendental for all formula_27.
The Riemann zeta function.
The Riemann zeta function 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 with real part greater than 1 by the following formula:
Taking , we see that 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 has a simple expression in terms of the Bernoulli numbers whenever is a positive even integer. With :
A rigorous proof using Euler's formula and L'Hôpital's rule.
The sinc function formula_32 has a Weierstrass factorization representation as infinite product:
The infinite product is analytic, so taking the natural logarithm of both sides and differentiating yields
After dividing the equation by formula_35 and regrouping one gets
We make a change of variables (formula_37):
Euler's formula can be used to deduce that
Then
Now we take the limit as formula_42 approaches zero and use L'Hôpital's rule thrice:
A rigorous proof using Fourier series.
Use Parseval's identity (applied to the function ) to obtain
where
for , and . Thus,
and
Therefore,
as required.
Another rigorous proof using Parseval's identity.
Given a complete orthonormal basis in the space formula_52 of L2 periodic functions over formula_53 (i.e., the subspace of square-integrable functions which are also periodic), denoted by formula_54, Parseval's identity tells us that
where formula_56 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 formula_58 such that formula_59. Then if we take formula_60, 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 relations.
Note that by considering higher-order powers of formula_63 we can use integration by parts to extend this method to enumerating formulas for formula_64 when formula_65. 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
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 the linked article.
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
between two expressions, each of which will tend to as 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 be a real number with , and let 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 , set , and consider for . Then is a multiple of and therefore . So,
where formula_74 and formula_75.