|講師：||HELMKE, Stefan 助教|
|題目：||The Basel Problem and the Riemann Hypothesis|
In the 14th century, the french philosopher
Nicole Oresme (c. 1325 - 1382) showed that the harmonic series
1 + 1/2 + 1/3 + 1/4 + 1/5 + ...
diverges. In contrast, the sum of the reciprocals of the squares of all natural numbers,
1 + 1/4 + 1/9 + 1/16 + 1/25 + ...
converges. In 1650, the Italian mathematician Pietro Mengoli (1626 - 1686) considered the problem to find the value of this sum and around the same time, in 1655, the English mathematician John Wallis (1616 - 1703) considered the same problem, probably independent of Mengoli. Since the sum converges very slowly it is difficult to even compute it only approximately. Many of the best mathematicians of that time, including Jacob (1654 - 1705) and Johann (1667 - 1748) Bernoulli, tried to solve this problem. But it was Leonhard Euler (1707 -1783) who succeeded in 1735 to find the exact answer. Nowadays, it is known as the Basel Problem, since so many mathematicians from the Swiss city of Basel were involved in its solution. But the story does not stop here. In fact it has merely begun. Euler continued his research on this series and already two years later, in 1737 he found a remarkable formula relating the series to prime numbers. In 1859, this formula would be the starting point of Bernhard Riemann's (1826 - 1866) analysis of his zeta function, in which he stated his famous conjecture, the Riemann Hypothesis, which is still unsolved and considered one of the most important problems in mathematics today.
In this lecture, I will discuss some of Euler's work related to the Basel Problem and give a short review of later developments.