[分析] 黎曼假設(閱讀數學科普書後的問題)
我現在在讀 質數魔力-橫跨兩世紀的狂熱 這本書
遇到一些問題要請教網友
∞ 1 1 1 1
ξ(s)=Σ --- = --- + --- + --- + ...
n=1n^s 1^s 2^s 3^s
在s>1的時候直接代入計算ξ(s)值
在s=1的時候發散
但接下來為了要計算0<s<1的ξ(s)值
∞ (-1)^(n-1) 1 1 1 1 1
書上定義η(s)=Σ ---------- = --- - --- + --- - --- + --- - ...
n=1 n^s 1^s 2^s 3^s 4^s 5^s
( 1 1 1 1 ) ( 1 1 1 ) ( 1 1 1 )
利用(--- - --- + --- - ---)=(--- + --- + --- ... ) - 2*(--- + --- + --- + ...)
(1^s 2^s 3^s 4^s) (1^s 2^s 3^s ) (2^s 4^s 6^s )
得到 η(s) = ξ(s) - 2^(1-s)*ξ(s)
ξ(s)=η(s)/(1-2^(1-s))
因為在0<s<1時η(s)是可以算的,再代入上面的式子得到ξ(s)值
但我的問題是明明在0<s<1時ξ(s)是發散的
結果利用恆等式硬是算出個值,但這根本就不合理阿
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