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 Ƃ̃e[}ƖړIF wWĂߒł́ARȊwA ЉȊwȂǂ̎X̊w╪ŒN邽߂ɁA ̐w̘gg݂ɂƂȂA VȊwIȕ@◝_ĂB ܂AtɁÂ悤ȐVꂪA ẘjSIȗ_ւƔWmꂸ݂B ̂悤ȐwƐ͂̓WJ̏ɂāǍ҂A ǧ𓥂܂IEIȍu`sB wE̖͂̌ʔE[A LȊwNɖĂ炤ƂӐ}ču`A Ƃė\m͉肵ȂB Q F QOQQNSPTij @@@@@@PUFST|PWFPT ꏊF SPPigcSفj utF Helmke, Stefan ځF Euler and the Basel Problem vF 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 [2]. A more comprehensive, but also more difficult account can be found in André Weil's book [1]. Also, still readable, is Euler's original book [3], first published in 1748 in Latin, but with many existing translations, including the English one below. Hiroshi Yuki's book [4] also contains an interesting chapter on the Basel Problem and the Euler Archive [5] has copies of most of Euler's original publications. André Weil, Number Theory; An approach through history; From Hammurapi to Legendre. Birkhäuser, 1983. (Chap. III.17-20.) Charles Edward Sandifer: The Early Mathematics of Leonhard Euler, The Mathematical Association of America, 2007. (Chap. 7, 21, 32 and 44.) Leonhard Euler: Introduction to analysis of the infinite, book I, translated by John D. Blanton, Springer, 1988. (Chap. 9-11.) _: wK[ \tgoN NGCeBu QOOVN (X) Online Resources: The Euler Archive (http://eulerarchive.maa.org/). "http://www.kurims.kyoto-u.ac.jp/ja/special-02.html"