I'm interested in the following zeta function:
$$\zeta_D(s)=\sum_{(m,n)\in \mathbb N^2} \frac{1}{(m^2+n^2)^s}$$
Where naturally $(m,n)\neq (0,0)$. I'm mainly interested in its poles (so I can perform an integral Mellin transform on it later), and so I'm looking for a way to simplify this expression, or write it in terms of other functions who's poles I know well.
I tried to somehow use the Hurwitz Zeta function, but I didn't get too far with this approach. Another approach I had in mind is using the Poisson summation formula, and identifying the terms in the sum as Fourier coefficients of some sort, but I didn't get too far with this either.
Does anyone have an idea of what I can do in order to find the poles and residues? Any help would be appreciated.
Thanks in advance.
There are two approaches,
one from the unique factorization in prime ideals in the PID $\Bbb{Z}[i]$ which leads to $$\sum_{(m,n)\ne (0,0)}(m^2+n^2)^{-s}=4\zeta(s)L(s,\chi_4)$$
and the other from $$\pi^{-s}\Gamma(s)\sum_{(m,n)\ne (0,0)}(m^2+n^2)^{-s}=\int_0^\infty (\theta(x)^2-1) x^{s-1}dx$$ where $\theta(x)=\sum_k e^{-\pi k^2 x}$ which is $=x^{-1/2}\theta(1/x)$ from the Poisson summation formula.
If you only care of the poles on $\Re(s)>1/2$ then the Gauss circle problem is the simplest solution.