Most general multiplicative sequence is ?

[benorin]

If $$a_{nm}=a_{n}a_{m}, \, \forall n,m\in\mathhbb{N} ,$$ then the sequence of complex terms $$a_{nm}$$ is (generally) of what form? That is, I would like to know what the most general sequence satisfying the above relation is. For example, it is clear that we must have $$a_{1}=1$$ and that for primes $$p_{k}$$ and integers $$\alpha_{k}$$ we have

$$a_{\prod p_{k}^{\alpha_{k}}} =\prod a_{p_{k}}^{\alpha_{k}}$$

and, clearly, for any constant b, the sequence $$a_{k}=k^{b}$$ is such a sequence, what other types of sequences qualify? would the lesser requirement that $$a_{2n}=a_{2}a_{n}, \, \forall n\in\mathhbb{N} ,$$ give any more possibilities?

 

[matt grime]

Surely you can give a complete classification of all possible sequences (what, again, constitutes 'generality' and how are you quantifying it?) since any such sequence is uniquely determined by its value at the prime indics, and any assignment of values to prime indices gives a sequence.

 

[benorin]

I just want an idea of what the typical families of sequences of this sort are.

 

[matt grime]

And I told you what every single sequence of this type is. They are called completely multiplicative functions.