The Basel problem is a mathematical problem that was unsolved for a long time and that was initially mainly dealt with by Basel mathematicians. It is a question of the sum of the reciprocal square numbers , i.e. the value of the series
It was solved in 1735 by Leonhard Euler, who found the series value .
In 1644 the Italian Pietro Mengoli asked himself whether this sum was converging, and if so, against what value, but could not answer this question. The Basel mathematician Jakob I Bernoulli found out about this problem a little later, but found no solution (1689). Several mathematicians then tried to answer the question, but were all unsuccessful. In 1726 Leonhard Euler , also a Basel mathematician and student of Jakob Bernoulli's brother Johann , began to deal with the problem. In 1735 he found the solution and published it in his work "De Summis Serierum Reciprocarum" .
and equated it with the product presentation of that function.
When (hypothetically) multiplying the infinite product, he only considered those products which contain and . Since there is no other way that a term can contain a quadratic term, the two quadratic terms on the respective sides must be the same.
and from this Euler deduced his solution:
About a double integral
The proof of a double integral appears as an exercise in William J. LeVeque's 1956 textbook on number theory. In it, he writes about the problem: “I have no idea where this problem came from, but I'm pretty sure it is with did not originate from me. "
The representation is first obtained
via the geometric row
Evaluating ζ (2) - 14 pieces of evidence for the value of ζ (2) compiled by Robin Chapman on a website of the University of Exeter (pdf, English; 181 kB)
literature
C. Edward Sandifer: Euler's solution of the Basel problem — the longer story. Euler at 300, 105-117, MAA Spectrum, Math. Assoc. America, Washington, DC, 2007.
Downey, Lawrence / Ong, Boon W. / Sellers, James A .: Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers. The College Mathematics Journal. Vol. 39, no. Nov. 5, 2008. P. 391-394