常用积性函数替换

$$
n = \sum_{d|n}\varphi(d)\ \ <==>\ \ id = \varphi*I
$$

$$
\frac{\varphi(n)}{n} =  \sum_{d|n}\frac{\mu(d)}{d}\ \ <==>\ \ \varphi = \mu*id
$$

$$
\varphi(n) =  \sum_{i=1}^{n}[\gcd(i, n)=1]
$$

$$
d(ij)=\sum_{x|i} \sum_{y|j}[\gcd(x, y)]
$$

$$
1 = \sum_{i|n}d(i)\mu(\frac{n}{i})\ \ <==>\ \ I=d*u
$$

$$
[n=1]=\sum_{d|n}\mu(d)\ \ <==>\ \ \xi=\mu*I
$$

$$
n = \sum_{i|n}\sigma(i)\mu(\frac{n}{i})\ \ <==>\ \ id=\sigma*\mu
$$