バーゼル問題の観察
ver 0.004 2022-10-6

更新履歴
ver 0.001 2022-10- 1 新規作成
ver 0.002 2022-10- 2 追記
ver 0.003 2022-10- 3 大幅改定
ver 0.004 2022-10- 4 挟み込みを考える　追加
ver 0.005 2022-10- 6 加算の工夫　追加

バーゼル問題とは

∞          1

∑　　----

n=1       n2

を求めることをバーゼル問題といいます。オイラーによって、無理数解が与えられていますが、有理数は四則演算で有理数で閉じているので、有理数解があるはず です。探してみましょう。

オ イラーの解法 加 算の順を工夫する 挟 み込みを考える
加算の工夫







有理数の和の一般解を考えてみましょう。ここでは、aj/bjは、aj=1、bjは=bj^2 です。ですからaj/bj=1/bj^2です。

したがって、分母だけの計算になります。つまり、結果は、l/mになります。分母mはすべての分母の積であり、分子lは、mに1/bjをかけ たものの総和ですから、l,mとも自然数ですから、有理数です。

2022-10-4追記

1πから7πまでの掛け算を計算してみました。
((1-x^2/(1π)^2) (1-x^2/(2π)^2) (1-x^2/(3π)^2) (1-x^2/(4π)^2) (1-x^2/(5π)^2) (1-x^2/(6π)^2) (1-x^2/(7π)^2))
=
x^14の項
- x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
x^12の項
+ x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
+ x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
+ x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
+ x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
+ x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2
+ x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2
+ x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2
 x^10の項
 - x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(7π)^2
 - x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2
 x^8の項
 + x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2 + x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 + x^2/(2π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 + x^2/(1π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(7π)^2 + x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2
 + x^2/(2π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2
 + x^2/(2π)^2 x^2/(3π)^2 x^2/(6π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(3π)^2 x^2/(6π)^2 x^2/(7π)^2
 + x^2/(1π)^2 x^2/(2π)^2 x^2/(6π)^2 x^2/(7π)^2 + x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2
 + x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2
 + x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(2π)^2 x^2/(5π)^2 x^2/(7π)^2
 + x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(7π)^2
 + x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(7π)^2
 + x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 + x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2
 + x^2/(1π)^2 x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2 + x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2
 + x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(6π)^2 + x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2
 + x^2/(1π)^2 x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2 + x^2/(1π)^2 x^2/(2π)^2 x^2/(5π)^2 x^2/(6π)^2
 + x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2 + x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(6π)^2
 + x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2 + x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2
 + x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 + x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2
 + x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2
 x^6の項
 - x^2/(4π)^2 x^2/(6π)^2 x^2/(7π)^2 - x^2/(5π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(4π)^2 x^2/(5π)^2 x^2/(7π)^2 - x^2/(3π)^2 x^2/(5π)^2 x^2/(7π)^2
 - x^2/(3π)^2 x^2/(4π)^2 x^2/(7π)^2 - x^2/(4π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(3π)^2 x^2/(6π)^2 x^2/(7π)^2 - x^2/(2π)^2 x^2/(6π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(6π)^2 x^2/(7π)^2 - x^2/(2π)^2 x^2/(5π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(5π)^2 x^2/(7π)^2 - x^2/(3π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(2π)^2 x^2/(4π)^2 x^2/(7π)^2 - x^2/(1π)^2 x^2/(4π)^2 x^2/(7π)^2
 - x^2/(2π)^2 x^2/(3π)^2 x^2/(7π)^2 - x^2/(1π)^2 x^2/(3π)^2 x^2/(7π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(7π)^2 - x^2/(2π)^2 x^2/(3π)^2 x^2/(6π)^2
 - x^2/(2π)^2 x^2/(5π)^2 x^2/(6π)^2 - x^2/(1π)^2 x^2/(5π)^2 x^2/(6π)^2
 - x^2/(3π)^2 x^2/(4π)^2 x^2/(6π)^2 - x^2/(2π)^2 x^2/(4π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(4π)^2 x^2/(6π)^2 - x^2/(1π)^2 x^2/(3π)^2 x^2/(6π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(6π)^2 - x^2/(3π)^2 x^2/(4π)^2 x^2/(5π)^2
 - x^2/(2π)^2 x^2/(4π)^2 x^2/(5π)^2 - x^2/(1π)^2 x^2/(4π)^2 x^2/(5π)^2
 - x^2/(2π)^2 x^2/(3π)^2 x^2/(5π)^2 - x^2/(1π)^2 x^2/(3π)^2 x^2/(5π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(5π)^2 - x^2/(2π)^2 x^2/(3π)^2 x^2/(4π)^2
 - x^2/(1π)^2 x^2/(3π)^2 x^2/(4π)^2 - x^2/(1π)^2 x^2/(2π)^2 x^2/(4π)^2
 - x^2/(1π)^2 x^2/(2π)^2 x^2/(3π)^2
 x^4の項 
 + x^2/(6π)^2 x^2/(7π)^2 + x^2/(5π)^2 x^2/(7π)^2 + x^2/(4π)^2 x^2/(7π)^2
 + x^2/(3π)^2 x^2/(7π)^2 + x^2/(2π)^2 x^2/(7π)^2 + x^2/(1π)^2 x^2/(7π)^2
 + x^2/(5π)^2 x^2/(6π)^2 + x^2/(4π)^2 x^2/(6π)^2 + x^2/(3π)^2 x^2/(6π)^2
 + x^2/(2π)^2 x^2/(6π)^2 + x^2/(1π)^2 x^2/(6π)^2 + x^2/(4π)^2 x^2/(5π)^2
 + x^2/(3π)^2 x^2/(5π)^2 + x^2/(2π)^2 x^2/(5π)^2 + x^2/(1π)^2 x^2/(5π)^2
 + x^2/(3π)^2 x^2/(4π)^2 + x^2/(2π)^2 x^2/(4π)^2 + x^2/(1π)^2 x^2/(4π)^2
 + x^2/(2π)^2 x^2/(3π)^2 + x^2/(1π)^2 x^2/(3π)^2 + x^2/(1π)^2 x^2/(2π)^2
x^2の項
 - x^2/(7π)^2 - x^2/(6π)^2 - x^2/(5π)^2 - x^2/(4π)^2 - x^2/(3π)^2 - x^2/(2π)^2 - x^2/(1π)^2
定数項
 + 1

x^2（実際はx^3)の項は、かんたんですが、x^4（実際はx^5)の項は、オイラーによるとπ^4/90だそうですが、これは、どうやって、求めたの だろうね？x^6の項は？・・・


メインページに戻る