We know that, given a lattice $\Lambda$ of the complex plane, the Eisenstein series $E_{2k} := \displaystyle\sum_{\substack{\omega \in \Lambda \\ \omega \neq (0,0)}} \frac{1}{|\omega|^{2k}}$ converges for $2k \geq 4$, i.e. for $k \geq 2$. However, I am wondering how $E_1(z)$ does not converge, which is the content of Stein and Shakarchi's complex analysis textbook question 3 from chapter 9, that $$\displaystyle\sum_{1 \leq n^2 + m^2 \leq R^2} \frac{1}{|n\tau + m|^2} = 2\pi \log R + O(1)$$.
I get the picture that we are trying to count all the lattices $\omega \in \Lambda$ inside a circle at the origin of radius $R$, and I also suspect that we are trying to use the fact that $\displaystyle\sum_{i = 1}^R \frac{1}{i}$ behaves like $\log R$ for $R$ large enough. However, I am having trouble coming up with a precise argument. On this document page 23, it says that the number of $\omega \in \Lambda$ such that $nr \leq |\omega| \leq nR$ is $4n$; why is that true?