Basel problem

It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734,[1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences.

[2] 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 solution to this problem can be used to estimate the probability that two large random numbers are coprime.

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

Euler assumed this as a heuristic for expanding an infinite degree polynomial in terms of its roots, but in fact it is not always true for general

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 [6]

Multiplying both sides of this equation by −π2 gives the sum of the reciprocals of the positive square integers.

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.

There are other forms of Newton's identities expressing the (finite) power sums

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

The zeta function is defined for any complex number s 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:

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.

The infinite product is analytic, so taking the natural logarithm of both sides and differentiating yields

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

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

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).

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,[11] 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".

[12] The main idea behind the proof is to bound the partial (finite) sums

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

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

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.

In van der Poorten's classic article chronicling Apéry's proof of the irrationality of