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} $