Are there any interesting (read: useful) extensions of the totient function $\phi(n)$ to work for all $n\in\mathbb{R}$ (or even $\mathbb{C}$)? Jordan's totient is almost this. I know this can maybe go in a group-theoretic direction (i.e. some kind of generalization of order in the multiplicitive group), but I was wondering if there's something simpler. As an example, it's well known that:

$$\sum_{n=1}^\infty \frac{\phi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}.$$

Is there an extension of this identity to:

$$\sum_{n=1}^\infty \frac{\phi(k n)}{n^s}=f(k,s),$$

where $k\in\mathbb{Z}$ and hopefully the r.h.s. is analytically extendable in $k,s$? It looks like you can split the left sum into terms with $(k,n)=1$ and otherwise. I'm not really seeing how to proceed, perhaps by applying mobius inversion to both sides?


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