|講師：||Helmke, Stefan 助教|
Euler and the Basel Problem
In the middle of the 17th century Pietro Mengoli,
successor of Bonaventura Cavalieri as professor of mathematics
at the University of Bologna, noticed that while the harmonic
series diverges, the sum of inverse squares is finite. But
in contrast to the other infinite sums he considered, he was
unable to compute this one, or even to find satisfying numerical
approximations, since it converges so slowly. Mengoli's work was
little known at his time, but other mathematicians independently
arrived at the same problem, with little more success though.
The development of the calculus partially improved the situation
so that by the beginning of the 1730's a few methods to compute
approximate values for the sum of inverse squares were known, one
of them due to Leonhard Euler. And then, in 1735, he suddenly
found the exact value! Though his argument was rather controversial
and it took him another 7 years before he had a satisfying proof
for the formula which led to his result, he did not stop here.
In this class we will follow Euler's ideas concerning the sum
of inverse squares, now known as the Basel Problem, supplemented
with some new insight and later developments.
References: An easily readable account on the Basel Problem is contained in Edward Sandifer's book . 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. Hiroshi Yuki's book  also contains an interesting chapter on the Basel Problem and the Euler Archive  has copies of most of Euler's original publications.