On the frequency of primes. Condsider the ("2D") sequence $(mn)_{m,n>1}$ (with $m,n \in \mathbb{N}$). It contains all natural numbers (in various multiplicities) except the primes and the number $1$. 
Now we construct a function with poles on the non-prime numbers
$$ N_s(x)=\sum_{m,n=2}^{\infty} (mn-x)^{-s}, $$ 
with $s>1$ and finite "gaps" exactly at the primes. In my understanding the fact that the function shows such "gaps" around all integer values $x$ only for primes and poles only around composite numbers, would qualify it for containing information on the frequencies of the primes, so to say in an 'inverse' manner. 
So my question is if one could extract informations on the prime number distributions from this function via Fourier analysis? In detail, if $N_s(x)$ is it well-defined, does it converge to a function with the outlined "gap property" and for which a FT can be found, and is it possible to find any (analytic) expression for this FT? Finally how does this FT look like?    

(I just assume (uniform) convergence of all expressions in the following)
The Fourier-transformation is linear, hence for $s\in\mathbb{N^+}$ it yields: 
$$\mathcal{F}\bigg[\sum_{m,n>1}^{\infty} (mn-x)^{-s}\bigg](t)  = \sum_{m,n>1}^{\infty} \mathcal{F}{[(mn-x)^{-s}]}(t) = \frac{2\pi}{(n-1)!} t^{n-1}\sum_{m,n>1}^{\infty} e^{i(mn)t} $$
So that means, in this way no more information on the distribution of the primes is yielded as compared to in the original expression (apart from that the series expression looks really worrisome in terms of convergence ...).
