Euler totient

Euler's totient function (sometimes called Euler's $\phi$ function) is the function

$\phi(n) =$ # of positive integers $\leq n$ that are relatively prime to $n$.

Properties

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 $n$.

Proof: proof goes here █

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 $n$.

Proof: proof goes here █