Theorem (1.19)

The summation of the Möbius Function over all $d|n$, where $n\in\mathbb{N}$ is $0$ except for $n=1$ $$\sum_{d|n}\mu(d)=\begin{cases} 1&n=1 \\ 0&\text{otherwise} \end{cases}$$

Theorem (Möbius Inversion Formula)

If $f,g$ are arithmetical functions then $$f(n)=\sum_{d|n}g(d)\iff g(n)=\sum_{d|n}\mu(d)f\left( \frac{n}{d} \right)$$

Theorem (Euler’s Totient Function in Terms of Mobius Function)

Euler’s Totient Function can be redefined in terms of the Möbius Function $$\frac{\phi(n)}{n}=\sum_{d|n}\frac{\mu(d)}{d}$$