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单词 ConvergenceOfRiemannZetaSeries
释义

convergence of Riemann zeta series


The series

n=11ns(1)

converges absolutely for all s with real partMathworldPlanetmath greater than 1.

Proof. Let  s=σ+it  where  σ and t arereal numbers and  σ>1.  Then

|1ns|=1|eslogn|=1eσlogn=1nσ.

Since the series  n=11nσ converges, by the p-test (http://planetmath.org/PTest), for  σ>1, we conclude that the series (1) is absolutely convergent in the half-plane  σ>1.

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更新时间:2025/5/4 17:34:45