|講師：||Helmke, Stefan 助教|
Euler and the Basel Problem
It had been known at least since the 14th century that
the harmonic series diverges. But in mid 17th century, it
had been observed that, in contrast, the sum of inverse squares
converges. However, it converges very slowly and it is therefore
rather difficult to compute. So to find the value of this sum
was one of the most challenging problems in mathematics of that
time. Many famous mathematicians tried to lay their hands on it
and failed, including the two Bernoulli brothers from Basel.
Then, with the appearance of Leonhard Euler, who also came
from Basel and was a student of Johann Bernoulli, the younger
of the two brothers, the situation unexpectedly changed and
Euler's name is inseparably connected with the solution of
this problem, which is thus known as the `Basel Problem'.
In my lecture, I will first briefly review the history of the problem (from Euler's perspective) and then explain his most important contributions to its solution. This will naturally lead to new, even much more interesting problems and I will conclude with a view into the future (again, from Euler's perspective), by shortly reviewing Bernhard Riemann's paper from 1859 on the distribution of prime numbers with its still unsolved `Riemann Hypothesis', one of the most challenging problems of our time!
References: An easily readable account on the Basel Problem is contained in Edward Sandifer's article  as well as in Raymond Ayoub's article . A more comprehensive, but also more difficult account can be found in André Weil's book . Also, still readable, is Euler's original book , first published in 1748 in Latin, but with many existing translations, including the English one below.