Show that $\sum\limits_{1\leq m^2+n^2 \leq R^2} \frac{1}{m^2+n^2} = 2\pi \log R +O(1)$ as $R\to \infty$

Question

This is an exercise in E.M.Stein's textbook: Complex Analysis P279-3. About proving the series $\sum_{n+m\tau \in \Lambda^{*}} \frac{1}{\vert n+m\tau \vert ^2}$ does not converge, where $\tau \in \mathbb{H}$ and lattices $\Lambda^{*}=\Lambda-(0,0)$. The idea is that we have known $\vert m\vert +\vert n\vert \approx \vert n+m\tau \vert.$ It suffices to show that

$$\sum_{1\leq m^2+n^2 \leq R^2} \frac{1}{m^2+n^2} = 2\pi \log R +O(1)$$ as $R\to \infty$. How to show the above identity?

Answer 1

HINT:

You can consider $a_{n} = \# \{(p,q)\| \ p^2 + q^2 = n\}$ and use $a_1 + \ldots + a_n \simeq \pi n $ to get an estimate for $\sum \frac{a_n}{n}$ using the Abel transformation.

If you only want the divergence of the series and not precise estimates, you can use $m^2 + n^2 \simeq (|m|+|n|)^2$ and $b_n= \#\{(p,q) | \ |p|+|q|=n\}= 4(n+1)$. In higher dimensions $d$, $b_n$ grows like $n^{d-1}$.

Answer 2

Hint: You can compare the sum with this integral

$$ \iint_{ 1 \leq x^2 + y^2 \leq R^2 } \frac{dx dy}{x^2 + y^2} = \int_1^R \int_0^{2\pi} r \frac{dr}{r^2} = 2\pi \log R, $$ where we used polar coordinates ( integration over spheres in higher dimensions) to compute the integral.