As a lover of number theory, the condition $f(xy)=f(x)f(y)$ in group morphisms always reminds me of completely multiplicative functions. Since the natural numbers $\mathbb{N}$ do not form a group under multiplication, we will work instead in the non-zero rational numbers $\mathbb{Q}^\*$ where the value of the multiplicative function at a rational number will be defined as


$$f\left(\frac{a}{b}\right)=\frac{f(a)}{f(b)}$$


which can be easily verified to be consistent under the multiplicativitiy hypothesis. This above representation means that we need to restrict ourselves to multiplicative functions that do not have $0$ under their image, but this isn't a huge loss. Collecting what we have above, every completely multiplicative function that does not have $0$ in its image induces a natural morphism $\mathbb{Q^\*}\to\mathbb{C}^\*$, and working backwards even morphism $\mathbb{Q^\*}\to\mathbb{C}^\*$ induces a natural multiplicative function $f$ that does not have $0$ in its image.


It feels very natural to me that the properties of morphisms $\mathbb{Q}^\*\to\mathbb{C}^\*$ have been studied extensively, and so I am wondering if there have been any papers published that utilize our knowledge about these morphisms to state new theorems about multiplicative functions.



Does viewing multiplicative functions as morphisms from $\mathbb{Q}^*\to\mathbb{C}^*$ have any consequnces?
Actually number theorists more-or-less do the *exact opposite* of what you suggest.


Namely, as explained in the comments, a multiplicative function is determined by its values on the primes. This suggest more that a multiplicative function should be thought of as a local object, rather than a global object.


So what normally happens is that one tries to come up with a $p$-adic analogue of the multiplicative function as well as a version at the infinite place, and then take a product over all places to obtain an adelic version
$$f\_{\mathbf{A}}: \mathbf{A}\_{\mathbb{Q}}^\times \to \mathbb{C}^\times$$
of the multiplicative function $f$, where $\mathbf{A}\_{\mathbb{Q}}$ denotes the ring of adeles of $\mathbb{Q}$ (restricted product over $\mathbb{R}$ and all $\mathbb{Q}\_p$).


Why is this the "opposite" of what you suggest? Well, my understanding is that there is no general method to construct $f\_\mathbf{A}$ from $f$. But one case where this is greatly studied is for *Dirichlet characters*. Here for a Dirichlet character $\chi$, the associated adelic version $\chi\_{\mathbf{A}}$ is *trivial* on $\mathbb{Q}^\times \subseteq \mathbf{A}\_{\mathbb{Q}}^\times$!


This is part of the general philosophy of automorphic forms. Namely that one studies adelic objects which are trivial on the corresponding rational objects, and this triviality usually has some interpretation in terms of a reciprocity law.


