Thinking along the lines of Tom Leinster's fascinating recent question, I'm wondering more generally about how to extend questions about natural numbers to groups, with the cyclic groups representing the natural numbers, and normal subgroups representing divisors. For example, let $\mathbb{G}$ be the set of all **isomorphism classes** of **finite** groups. Then an "arithmetic function" could be defined to be simply a function $f:\mathbb{G}\rightarrow\mathbb{C}$. Here are a few analogs I wrote down rather quickly (so perhaps there are better proposals for generalizations than these):
$$\text{id}(G)=|G|\quad\quad\quad \epsilon(G)=\begin{cases}1\text{ if $G$ is trivial}\\\ 0\text{ otherwise}\end{cases}\quad\quad\quad z(G)=0$$
$$\sigma_k(G)=\sum_{N\\,\triangleleft \\,G} |N|^k \quad\quad\quad\quad\phi(G)=|G|\prod_{\substack{N\\,\triangleleft \\,G \\\ |N|\text{ prime}}}\left(1-\frac{1}{|N|}\right)$$
Tom Leinster's question is whether there is a solution to $\sigma_1(G)=2|G|$.

Question 1: What is known, if anything, about these functions? What is a good proposal for an analog of the Mobius function (I couldn't think of one offhand)? Can anyone demonstrate that they satisfy formulas that are analogs of their natural-number counterparts, or perhaps instead give examples that would indicate that these functions act weird and aren't nice generalizations to make?

I suspect that if these functions act badly, the most likely fix would be to redefine $\mathbb{G}$ to be isomorphism classes of finite **abelian** groups. All subgroups are normal, and the structure theorem makes things much more controlled. I would guess that defining (for example) what it means for two groups to be "coprime", and hence what it means for an "arithmetic function" to be "multiplicative", would go over much more smoothly with abelian groups.

Going further: given two "arithmetic functions" $f,g:\mathbb{G}\rightarrow\mathbb{C}$, we can define a "Dirichlet convolution" by $$(f\ast g)(G)=\sum_{N\\,\triangleleft \\,G}f(N)g(G/N)$$ I've got to say, my jaw dropped a bit when I wrote that down. But one immediate difference I can see, somewhat discouraging, is that $\ast$ would not be abelian, since we aren't guaranteed that there are any normal subgroups $M\triangleleft G$ isomorphic to $G/N$ (see this MO question), much less that if $M\cong\\! G/N$ then $G/M\cong\\!\\! N$. However, the function $\epsilon$ is still a left and right identity for $\ast$, and $z$ is still a zero.

Question 2: Can anyone prove or disprove that $\ast$ is associative? If it is, we at least get a non-commutative ring under $\ast$ and pointwise addition (that $\ast$ distributes over pointwise sums is obvious).

Now I suspect I am really getting "greedy" with my generalizing. We could further define "Dirichlet series", such as the zeta function (note that this is *not* the same thing as the zeta function of a group):
$$\zeta_{\mathbb{G}}(s)=\sum_{G\in\mathbb{G}}\frac{1}{|G|^s}=\lim_{n\rightarrow\infty}\sum_{\substack{G\in\mathbb{G}\\\ |G|\leq n}}\frac{1}{|G|^s}$$
I seriously doubt there is any hope for an Euler product-like expression. But perhaps, if we restricted ourselves to finite abelian groups...

Also, I'm familiar with the result that the number of isomorphism classes of groups of order $p^n$ grows as $p^{\frac{2}{27}n^3+O(n^{8/3})}$, and I imagine the growth rate is at least as bad for not-necessarily $p$-groups.

Question 3: Is there any $s>0$ for which $\zeta_{\mathbb{G}}(s)$ converges?

I wrote this question rather fast, and I welcome any feedback on how to improve it, or make it more appropriate for MO. Should I break this up into multiple questions? Is it too open-ended?