Value of Multiplicative Function at One


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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 $\map f 1 = 1$.

$\blacksquare$

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