Theorem: The function $\phi$ obeys the formula $$\phi(n) = \displaystyle\sum_{d|n} \mu(d) \dfrac{n}{d},$$ where the notation $d | n$ indicates that $d$ is a divisor of $n$ and $\mu$ is the Möbius function.

Theorem: The function $\phi$ obeys the formula $$\phi(n) = n \displaystyle\prod_{p | n} \left( 1 - \dfrac{1}{p} \right),$$ where the notation $p | n$ indicates that $p$ is a prime that divides $n$.

Theorem: The following formula holds: $$\phi(n) = n\lim_{s \rightarrow 1} \zeta(s) \displaystyle\sum_{d | n} \mu(d)(e^{\frac{1}{d}})^{s-1},$$ where $\zeta$ is the Riemann zeta function and $\mu$ is the Möbius function, $e$ is the base of the exponential and the notation $d|n$ indicates that $d$ is any divisor of $n$.