欧拉函数

\[n = \displaystyle\sum_{d \mid n} \varphi(d) \]

证明：

\[\forall n \in \mathbb{N}_+, f(x) = \displaystyle\sum_{i = 1}^n [\gcd(i, n) = x] \]

\[\because \gcd(i, n) = x \]

\[\therefore \gcd(\displaystyle\frac{i}{x}, \displaystyle\frac{n}{x}) = 1 \]

\[\therefore f(x) = \varphi(\displaystyle\frac{n}{x}) \]

\[\because \forall i \in [1, n] \bigcap Z, 1 \leq \gcd(i, n) \leq n \]

\[\therefore n = \displaystyle\sum_{i = 1}^{n} f(x) \]

\[\because \forall i \nmid n, f(i) = 0 \]

\[\therefore n = \displaystyle\sum_{d \mid n} f(d) = \displaystyle\sum_{d \mid n} \varphi(\displaystyle\frac{n}{d}) = \displaystyle\sum_{d \mid n} \varphi(d) \]

证毕．