Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits\_{(m,n) \in D\_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D\_r$ denotes the closed disk of radius $r$ centered at the origin. 


I expect one would need Poisson summation to turn this into an exponential sum and apply some analytic estimate, but I cannot find any relevant results. Any help and especially, references, would be welcome.



An interesting sum over lattice points in a large disk centered at the origin
It is problem number 10 of IMC 2018, you may find the solution on the [official site](http://www.imc-math.org.uk/?year=2018&section=problems&item=prob10q).


