Question:
Exercise 1.1.4(a)Let $k\geq 3$ be an integer and let $L' = \mathbb{Z}^2 \setminus \{(0,0)\}$. Show that the series $$\sum_{(c,d) \in L'} (\sup\{|c|, |d|\})^{-k}$$ converges by considering the partial sums over expanding squares.
Can someone explain what does it mean by 'expanding squares'? Or any other hints to prove the convergence?
Source: Diamond and Shurman's A First Course of Modular Forms.
My attempt following the hints provided above:
Let $R >0$ (for convenience we set $R$ to be an integer) be given and by symmetry the partial sum $$S_R := \sum_{\substack{- R \leq c \leq R\\ -R \leq d \leq R}} (\sup\{|c|, |d|\})^{-k}$$ can be written as $4$ times of the sum in the first quadrant, more specifically $$S_R =4 S_1 := 4 \sum_{\substack{0 \leq c \leq R\\ 0 \leq d \leq R\\ c^2 + d^2 \ne0}} (\sup\{|c|, |d|\})^{-k}$$ By symmetry with respect to the line $y=x$ we have $$S_1 := \sum_{n=1}^R \frac{1}{n^k} + 2 \sum_{0 \leq d \leq R} \sum_{d < c \leq R} \frac{1}{c^k}$$ where the double sum on the right can be evaluated as \begin{align*} \sum_{0 \leq d \leq R} \sum_{d < c \leq R} \frac{1}{c^k} &= \sum_{n=1}^R \frac{1}{n^k} + \sum_{n=2}^R \frac{1}{n^k} + \dots + \sum_{n=R-1}^R \frac{1}{n^k}\\ &= \frac{1}{1^k} + 2 \frac{1}{2^k} + \dots + R \frac{1}{R^k}\\ &= \frac{1}{1^{k-1}} + \frac{1}{2^{k-1}} + \dots + \frac{1}{R^{k-1}}\\ &= \sum_{n=1}^R \frac{1}{n^{k-1}} \end{align*} Thus, the initial partial sum is bounded above by $$4\left(\sum_{n=1}^R \frac{1}{n^k} + 2\sum_{n=1}^R \frac{1}{n^{k-1}} \right)$$ which is certainly bounded as $R$ tends to infinity since $k \geq 3$.
Do let me know if there are any gaps in the proof.