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Wikipedia / Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was 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 more than a century 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 the city of 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 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 , the reciprocal of the solution to the Basel problem.

Euler's approach

essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series.

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

The Weierstrass factorization theorem shows that the right-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.

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

Multiplying both sides of this equation by −2 gives the sum of the reciprocals of the positive square integers.

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

. Then using known formulas for elementary symmetric polynomialss (a.k.a., Newton's formulas expanded in terms of power sum

. There are other forms of Newton's identities in terms of the elementary symmetric polynomials 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

which in our situation equates to the limiting recurrence relation (or generating function convolution, or product

Consequences of Euler's proof

is always a rational and integer powers of it are transcendental is irrational, and more precisely, transcendental. By contrast, the properties of the odd-indexed zeta constants, including Apéry's constant, are almost completely unknown.

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

Taking , we see that

Convergence can be proven by the integral test

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

The normalized sinc function has a Weierstrass factorization

The infinite product is analytic, so taking the natural logarithm

(by uniform convergence

Euler's formula or using the corresponding hyperbolic function

Now we take the limit approaches zero and use L'Hôpital's rule thrice. By Tannery's theorem, we can interchange the limit and infinite series

A proof using Fourier series

Use Parseval's identity (applied to the function

for , and

as required.

Another proof using Parseval's identity

Given a complete orthonormal basis of L2 periodic functions (i.e., the subspace of square-integrable functions which are also periodic, Parseval's identity

is defined in terms of the inner product on this Hilbert space

We can consider the orthonormal basis

by elementary calculus and integration by parts, respectively. Finally, by Parseval's identity

Generalizations and recurrence relations

we can use integration by parts

so that integration by parts yields the recurrence relation

Then by applying Parseval's identity as we did for the first case above along with the linearity of the inner product

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:

While the primitive function we arrive at

which can be integrated by substituting the definite integral reduces to

The expression can be simplified using the arctangent addition formula by means of trigonometric substitution, resulting in

The integration constant are related by

we can factor using the logarithm of a power identity and the substitution, and it follows that

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

The last two identities imply

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

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

From the binomial theorem

We take this identity, fix a positive integer , set , and consider for . Then is a multiple of and therefore

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

By Vieta's formulas

Substituting the identity

Now consider the inequality (illustrated geometrically above). If we add up all these inequalities for each of the numbers

Multiplying through by

As approaches infinity, the left and right hand expressions each approach , so by the squeeze theorem

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

. An invariant volume form

is the p-adic integers.

is the congruence subgroup is the normalized Haar measure.

At the infinite place, an integral computation over the fundamental domain On the right-hand side, we recognize the Euler product, and so this gives the solution to the Basel problem.

.

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.

  • equally spaced points on a circle with circumference and the length of each arc.

  • without loss of generality).

  • Draw all the chords points. Now (this is the key to the proof), compute the sum of the inverse squares.

  • ), . increases, the number of chords increases, but their length increases too (as the circle gets bigger), so their inverse square decreases.

  • .

  • are positioned at the odd, and 2 onward. You hence get this variation of the Basel Problem:

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

that is,

or

.

, which still allows the limiting argument to be applied. The proof proceeds by induction, and uses the Inverse Pythagorean Theorem, which states that:

is the height of a right triangle.

  • as stated above.

  • .

  • is always even, and a simple geometric argument shows that you can pick pairs) and the other in the "upper" half.

. You can then make the following considerations:

  • is a diameter.

  • is a diameter.

  • for the Inscribed Angle Theorem.

  • , again because the radius is half.

  • , hence for the Inverse Pythagorean Theorem:

  • .

  • 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 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

Continued fractions

In van der Poorten's classic article chronicling \zeta(3), the author notes as "a red herring" the similarity of a simple continued fraction. Another continued fraction of a similar form is:.

See also

References

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Notes