In https://math.mit.edu/classes/18.785/2015fa/LectureNotes17.pdf, just after Definition 17.4 on page 7, the author states that "Every $m$-periodic Dirichlet character $\chi$ restricts to a group character on $(\mathbb{Z}/m\mathbb{Z})^\times$; conversely, every group character $\chi$ of $(\mathbb{Z}/m\mathbb{Z})^\times$ can be extended to a Dirichlet character $\chi$ by defining $\chi(n)=0$ for $n\not\in(\mathbb{Z}/m\mathbb{Z})^\times$." In Remark 17.5 on the same page the author clarifies what is meant by $n\not\in(\mathbb{Z}/m\mathbb{Z})^\times$ - it means the image of $n$ under the quotient map $\mathbb{Z}\to\mathbb{Z}/m\mathbb{Z}$ is not in $(\mathbb{Z}/m\mathbb{Z})^\times$.
But what confused me is that, in Definition 17.4, the author defines a Dirichlet character to be simply a period, completely multiplicative arithmetic function. So why is it that a Dirichlet character can only be viewed as an extension of a character of $(\mathbb{Z}/m\mathbb{Z})^\times$, as opposed to an extension of a character simply of $\mathbb{Z}/m\mathbb{Z}$?
I have thought about this all day, and attempted to find a proof as to why $(\mathbb{Z}/m\mathbb{Z})^\times$ rather than $\mathbb{Z}/m\mathbb{Z}$ is needed, but have been unsuccessful. What I'd really like to see is a concrete example that shows how Definition 17.4 would NOT be satisfied if $\chi$ were an extension of $\mathbb{Z}/m\mathbb{Z}$ rather than $(\mathbb{Z}/m\mathbb{Z})^\times$.
Any help is greatly appreciated!
$\mathbf Z/ m\mathbf Z$ is a group under addition with $m$ elements, so a group homomorphism $$\mathbf Z/ m\mathbf Z \to \mathbf C^\times$$ would not give a multiplicative function when viewed as a function from $\mathbf Z\to \mathbf C$, it would be an additive function instead.
To give an example we have a character $\mathbf Z/2\mathbf Z \to \mathbf C^\times$ given by $1\mapsto -1$ and $0\mapsto 1$, here this is no longer a multiplicative function when viewed as a function $$\chi:\mathbf Z \to \mathbf C^\times$$ as $-1= \chi(1) \ne \chi (1) \cdot \chi(1) = 1$.