The Basel Problem is named for the city of Basel in Switzerland where two of the Bernoulli brothers, Jakob Bernoulli and Johann Bernoulli both served as professor of mathematics at the university there. Jakob Bernoulli stated the problem, and it became known as the Basel problem ever since.

As Jakob Bernoulli found a proof of the divergence of the harmonic series (as had his brother Johann), he then wondered what would happen if every term of the harmonic series were squared. It seemed that the harmonic series just barely diverged, so it might be that this new series derived from it (which came to be known as the Basel series) converged, and if it did, what was its value. So, in 1689 Bernoulli posed the Basel problem, which was to find a closed form for the infinite series:

     1     1     1
1 +  __ +  __ +  __ + ...
      2     2     2
     2     3     4

A solution eluded everyone, including the great Gottfried Leibniz, until the problem was finally cracked 46 years later, by the young Leonhard Euler in St. Petersburg who showed that the Basel series was convergent, and it converged to the value π2/6. This was a very striking result for its time, in that the number π appeared in a context that had nothing whatsoever to do with circles or geometry, the domains in which π was assumed at the time to exclusively be relevant.

His proof is quite ingenious, and uses some pretty neat relations between infinite series and infinite products. He began by taking the sinc function sin(x)/x, which has the Maclaurin series:

1 - x2/3! + x4/5! - x6/7! ...

He then noted that sinc(x) = sin(x)/x has roots at x = ±kπ, for k = 1, 2, 3, ..., so sinc(x) may be expressed as the infinite product:

sinc(x) = (1 - x/π)(1 + x/π)(1 - x/2π)(1 + x/2π)(1 - x/3π)(1 + x/3π)...

which may also be written by combining the differences of two squares evident there, as:

sinc(x) = (1 - x22) (1 - x2/4π2) (1 - x2/9π2) (1 - x2/16π2)...

Expanding this series and collecting similar terms yields this infinite series of infinite series:

sinc(x) = 1 - (1/π2 + 1/4π2 + 1/9π2 + ...)x2 + (1/4π4 + 1/9π2 + 1/16π2 +...)x4 - ...

Then Euler equated the original Maclaurin series for sinc(x) and his new series, giving:

1 - x2/3! + x4/5! - x6/7! ... = 1 - (1/π2 + 1/4π2 + 1/9π2 + ...)x2 + (1/4π4 + 1/9π2 + 1/16π2 +...)x4 - ...

For this equation to be true, he realized that the corresponding coefficients must also be equal, and in particular, it must be true that:

-1/3! = - (1/π2 + 1/4π2 + 1/9π2 + ...)

so

1/6 = (1 + 1/4 + 1/9 + ...)1/π2

finally leading to the dramatic result:

π2/6 = 1 + 1/22 + 1/32 + 1/42 + ..., voila, the Basel series!

Euler actually did one better and managed to find closed forms for all series of the form:

            1     1     1
ζ(n) = 1 +  __ +  __ +  __ + ...
             n     n     n
            2     3     4

(which we would today call the Riemann zeta function), for all even values of n. When n is 4, he found it to converge to π4/90, and &zeta(6) = π6/945, and he went on and on until n = 26, where the zeta function converges to 1,315,862π26/11,094,481,976,030,578,125. In general, he found that ζ(2k) for k a positive integer is given by the formula: (-1)k-1B2k(2π)2k/2(2k)!, where the B2k are the Bernoulli numbers.

But what happens when n is, say 3, or some other odd number greater than 1? Euler's result has nothing to say about that, and the 260 years since have yielded few positive results. No one knows as of this writing whether a closed form even exists. Srinivasa Ramanujan did a lot of interesting work on this problem in the early 20th century, however it was only in 1978 that Roger Apéry managed to prove that ζ(3) was actually irrational, and its value has come to be known as Apéry's Constant. An extension of Apéry's result to other odd values remains, in the words of Alf van der Poorten: "a mystery wrapped in an enigma." Further, it is still unknown whether ζ(3) is transcendental, or even whether ζ(3)/π3 is irrational.

Sources

John Derbyshire, Prime Obsession, chapter 5.

Apéry's Constant from MathSoft, an archived page resurrected by the Wayback Machine: http://web.archive.org/web/20001205115300/http://www.mathsoft.com/asolve/constant/apery/apery.html

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