Dilation Invariance

Given the formula $$F(x)= \sum_{n=-\infty}^{\infty}f(x+n)$$ We know that is invariant under translations of the form $y=x+n$ for any integer $n$. However can we find a similar formula for dilations ??

I believe that $H(x)= \sum_{Q}f(qx)$ is invariant under dilations of the form $y=nx$ for integer $n$, where $\mathbb{ Q}$ is the set of rational numbers.

In this last case , what would be the Mellin transform of $H(x)$? I believe that it will be $F(s)\zeta(s) \zeta (-s)$, but I am not sure , here $F(s)= \int_{0}^{\infty}dtH(t)t^{s-1}$

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Can the sum converge? Continuity of $f$ at $x=a$ and $f(a)\neq 0$ seems to imply that the sum diverges at that point. –  sos440 Mar 30 '12 at 9:00
oK, let us suppose the SUm is ALWAYS convergent and also $f(0)=0$ and $\int_{0}^{\infty}f(t)dt =0$ –  Jose Garcia Mar 30 '12 at 9:47
I think that there's little hope that the sum $\sum_{\Bbb Q}$ converges unless $\Bbb Q$ is discrete. This is realized, for instance, in the embedding of $\Bbb Q$ in its ring of adeles $\Bbb A=\Bbb Q\Bbb R\hat{\Bbb Z}$. –  Andrea Mori Mar 30 '12 at 9:50
for example , if we use the fundamental theorem of arithmetic i guess thati could write $H(x)= \sum_{m=-\infty}^{\infty}\sum_{p}f(p^{m})$ so it involves a sum over primes and prime powers –  Jose Garcia Mar 31 '12 at 12:43