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, 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, and it was not until 1741 that he was able to produce a truly rigorous proof.

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, 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 left-hand side is the product of linear factors given by its roots, just as we do for finite polynomials :
If we formally multiply out this product and collect all the terms, 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 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 formulas obtained from elementary symmetric polynomials, this same approach can be used to enumerate formulas 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 expanded as above be defined by. Then using known formulas for elementary symmetric polynomials, we can see that
and so on for subsequent coefficients of. There are other forms of Newton's identities expressing the power sums in terms of the elementary symmetric polynomials, but we can go a more direct route to expressing non-recursive formulas for 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 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 explained above and the extension of his method by elementary symmetric polynomials in the previous subsection, we can conclude that is always a rational multiple of. 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, we can conclude much more about this class of zeta constants. In particular, since and integer powers of it are transcendental, we can conclude at this point that is irrational, and more precisely, transcendental for all.

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 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 and regrouping one gets
We make a change of variables :
Euler's formula can be used to deduce that
Then
Now we take the limit as approaches zero and use L'Hôpital's rule thrice:

A rigorous proof using Fourier series

Use Parseval's identity 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 of L2 periodic functions over , denoted by, Parseval's identity tells us that
where 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 such that. Then if we take, 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 we can use integration by parts to extend this method to enumerating formulas for when. 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.
For a proof using the residue theorem, see the linked article.

History of this proof

The proof goes back to Augustin Louis Cauchy. 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 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,
for every. The values are distinct numbers in the interval. Since the function is one-to-one on this interval, the numbers are distinct for. By the above equation, these numbers are the roots of the th 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, we have
Now consider the inequality . If we add up all these inequalities for each of the numbers, and if we use the two identities above, we get
Multiplying through by, this becomes
As approaches infinity, the left and right hand expressions each approach, so by the squeeze theorem,
and this completes the proof.

Other identities

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.

Series representations

The following are series representations of the constant:
There are also BBP-type series expansions for.

Integral representations

The following are integral representations of

Continued fractions

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
and
where and.