几个常用的莫比乌斯反演公式(持续更新)
前言
定义一下几个常用函数。
\(I(n) = 1\)
\(N(n) = n\)
\(u(n) = \lfloor \frac{1}{n} \rfloor\)
\(n\) = \(p_1^{a_1} \times p_2^{a_2} \times ... p_r^{a_r}\) 当有任意\(a_i\)>\(1\)时：\(\mu(n) = 0\) 否则\(\mu(n) = (-1)^r\)
\(u\ * \ \mu = I\)
正文
\(No.1\) \(\sum_{d|n} \varphi(d) = n\)
引导证明（I）(需要用到\(No.2\)与\(No.3\))
\(\sum_{d|n} \varphi(d) = n\)
\(\sum_{d|n} \varphi(d)\ * \ u(d) = n\)
\(\varphi\ * \ u= N\)
\(\varphi\ * \ u\ * \ \mu = N \ * \ \mu\)
\(\varphi = N \ * \ \mu\)
\(\varphi(n) = \sum_{d|n}\mu(d)\frac{n}{d}\)
\(No.2\) \(N * M = \sum_{d|n}N(d)*M(\frac{n}{d})\) (狄利克雷卷积)
\(No.3\) 如果存在\(F(n)=\sum_{d|n}f(d)\)
可以推出\(f(n)=\sum_{d|n}F(d)\mu(\frac{n}{d})\) (莫比乌斯反演)
