狄利克雷卷积 一些常识

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$\large e=\mu*I$
$\large \varphi*I=id\\\to\varphi=\mu*id$
$\large \sigma_0=I*I\\\to I=\mu*\sigma_0$

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$\large \sum_{d|n}\mu(d)=[n=1]$
$\large \sum_{d|n}\varphi(d)=n$
$\large \sum_{d|n}\mu(\frac nd)\cdot d=\varphi(n)\\\sum_{d|n}\mu(d)\cdot \frac nd=\varphi(n)$
$\large \sigma_0(i\cdot j)=\sum_{x|i}\sum_{y|j}[(x,y)=1]$
$\large \sigma_1(i\cdot j)=\sum_{x|i}\sum_{y|j}x\cdot j/y[(x,y)=1]$