Looking for an infinite series resembling the exponential function

I'm looking for a function $$f(x)$$ that has the following property:

$$\sum_{x=1}^\infty f(kx) = r^k$$

for some real $$0 < r < 1$$, and at least for strictly positive integer $$k$$.

Does such a function exist?

This could also be thought of in terms of some sequence of real numbers $$f[n]$$.

• Do you really want a function $\;f\;$ such that $\;f(k)+f(2k)+f(3k)+\ldots=r^k\;$ ? – DonAntonio Jan 19 at 0:06
• Right. Is there something wrong with that? – Mike Battaglia Jan 19 at 0:07
• @Mi No, not really...It is just that it looks pretty weird to me, that's all. It isn't a power series nor a series of functions, but just the sum of the values of $\;f\;$ on integral multiples of some number $\;k\;$ ....and that must equal $\;r^k\;$ ...It'll be interesting if someone can come up with something like that. – DonAntonio Jan 19 at 0:09
• That's true. Would it be clearer if I rewrote this in terms of some sequence f[n] instead of f(x)? – Mike Battaglia Jan 19 at 0:10
• Now posted to MO, mathoverflow.net/questions/321432/… – Gerry Myerson Jan 23 at 20:00

We have that $$\;r^k=e^{k\log r}\;$$ , so
$$e^{k\log r}=\sum_{n=0}^\infty\frac{k^n(\log r)^n}{n!}$$