# 大学院教育・講義

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## 全学共通科目講義（１回生～４回生対象）

 現代の数学と数理解析 ――　基礎概念とその諸科学への広がり
 授業のテーマと目的： 数学が発展してきた過程では、自然科学、 社会科学などの種々の学問分野で提起される問題を解決するために、 既存の数学の枠組みにとらわれない、 新しい数理科学的な方法や理論が導入されてきた。 また、逆に、そのような新しい流れが、 数学の核心的な理論へと発展した例も数知れず存在する。 このような数学と数理解析の展開の諸相について、第一線の研究者が、 自身の研究を踏まえた入門的・解説的な講義を行う。 数学・数理解析の研究の面白さ・深さを、 感性豊かな学生諸君に味わってもらうことを意図して講義し、 原則として予備知識は仮定しない。 第８回 日時： ２０１５年６月５日（金） １６：３０－１８：００ 場所： 数理解析研究所　４２０号室 講師： HELMKE, Stefan 助教 題目： The Basel Problem and the Riemann Hypothesis 要約： 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. References: Raymond Ayoub, Euler and the Zeta Function, American Mathematical Monthly, Vol. 81 (1974), 1067--1086. This article is reprinted in The Genius of Euler, Reflections on his Life and Work, William Dunham (Editor), The Mathematical Association of America, 2007. C. Edward Sandifer, The Early Mathematics of Leonard Euler, The Mathematical Association of America, 2007. In particular Chapters 7, 21 and 32. Raghavan Narasimhan, Editor's Preface (Together with a mathematical commentary on some of Riemann's work) in B. Riemann, Gesammelte Mathematische Werke, Wissenschaftlicher Nachlass und Nachträge, Springer, 1990. "http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"

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