IMC 2018 Problem 10

For $R>1$ let $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$ . Compute $$\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.$$ Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro