Value of Multiplicative Function at One
Jump to navigation
Jump to search
This article needs to be linked to other articles. In particular: Identically zero You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
Theorem
Let $f: \N \to \C$ be a multiplicative function.
If $f$ is not identically zero, then $\map f 1 = 1$.
Proof
If $f$ is not identically zero, then:
- $\exists m \in \Z: \map f m \ne 0$
Then:
- $\map f m = \map f {1 \times m} = \map f 1 \, \map f m$
Hence $\map f 1 = 1$.
$\blacksquare$