International Mathematics Competition
for University Students
2018

IMC2018: Day 2, Problem 10

Problem 10. For \(\displaystyle R>1\) let \(\displaystyle \mathcal{D}_R = \{(a,b)\in\mathbb{Z}^2 \colon 0<a^2+b^2<R\}\). Compute

\(\displaystyle \lim_{R\rightarrow \infty} \sum_{(a,b) \in \mathcal{D}_R} \frac{(-1)^{a+b}}{a^2+b^2}.\)

(Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro)

Hint: Take the difference between two sums with all positive signs.

  Solution