Consider the following summation:
$$\sum_{n,d} \frac{δ\left(t - \tfrac{n}{d}\right)}{(nd)^σ}$$
Where $δ(t)$ is the Dirac delta distribution, and $σ$ is a constant, and $n,d>0$.
Every partial sum exists. It is easy to come up with a sequence whose limit would be the infinite summation I want over all n,d. But does the limit actually exist as a distribution, at least for some choice of σ?
Furthermore, supposing it exists, does the integral of this distribution exist? If so, is there any expression for it?
I came up with this while messing around with the zeta function, and hit a sticking point here.
Take a test function $\phi$ with support in $[-R,R]$. Then your operator applied to $\phi$ becomes
$$\sum _{n,d}\frac{\phi(n/d)}{(nd)^\sigma}\le \|\phi\|_{\infty}\sum_{n,d:1\le n\le Rd}\frac{1}{(nd)^\sigma}.$$
It easy to check that the sum $\sum_{n,d:1\le n\le Rd}\frac{1}{(nd)^\sigma}$ is finite for $\sigma>1$. Hence, by definition, your operator is a distribution of order zero.