Why is the inner sum of $ \sum_t |z_t|^2 \sum_r \frac{1}{(\lambda_r-\lambda_t)^2} $ bounded by $ 2\delta^{-2}\zeta(2) = \frac{\pi^2}{3\delta^2}$? Let $\lambda_r$ be distinct real numbers with $|\lambda_r- \lambda_t| \geq \delta$ if $r \neq s$. Why is the inner sum of 
$$
\sum_t |z_t|^2 \sum_r \frac{1}{(\lambda_r-\lambda_t)^2}
$$
bounded by
$$
2\delta^{-2}\zeta(2) = \frac{\pi^2}{3\delta^2}.
$$
I can't really see this. I tried using the assumption but it led me nowhere.
 A: Define $\mu_r=\lambda_r-\lambda_t$. Observe that for $r\neq s$ we still have $$|\mu_r-\mu_s|=|\lambda_r-\lambda_s|\geq\delta$$ and we are trying to estimate the sum $$\sum_{r\neq t}\frac1{\mu_r^2}.$$ Now, partition the real line $\mathbb R$ into intervals $$\mathbb R=\bigcup_{k\in\mathbb Z}J_k,$$ where $J_k$ are defined as follows ($k\in\mathbb Z$): $$J_k=\begin{cases}(-\delta,\delta);&k=0,\\ [k\delta,k\delta+\delta),&k>0,\\(k\delta-\delta,k\delta]&k<0.\end{cases}$$ Observe that each interval $J_k$ contains at most one number $\mu_r$. For $k\neq 0$, this is because for any $x,y\in J_k$ we have $|x-y|<\delta$, whereas for $r\neq s$ we have $|\mu_r-\mu_s|\geq\delta$. For $k=0$, we have $0=\mu_t\in J_0$, so any $x\in J_0$ must satisfy $|x-\mu_t|<\delta$, whereas for $r\neq t$, $|\mu_r-\mu_t|\geq\delta$.
Now define $$\nu_k=\begin{cases}\mu_r;&\text{if }\mu_r\in J_k,\\k\delta;&\text{otherwise.}\end{cases}$$ In particular, we have $$\{\mu_r|r\neq t\}\subseteq\{\nu_k\mid k\in\mathbb Z\setminus\{0\}\}.$$ Therefore, $$\sum_{r\neq t}\frac1{\mu_r^2}\leq\sum_{k\neq 0}\frac1{\nu_k^2}\leq\sum_{k\neq 0}\frac1{(k\delta)^2}=\frac2{\delta^2}\sum_{k=1}^{\infty}\frac1{k^2}=\frac2{\delta^2}\frac{\pi^2}6,$$ where the second inequality follows from the fact that $\nu_k\in J_k$.

The intuitive explanation is as follows: any two numbers $\lambda_r$ are at least $\delta$ apart. In the worst case (largest contribution to the sum) two consecutive $\lambda_r$ are exactly $\delta$ apart and the numbers $\lambda_r$ extend towards both ends of the real line, i.e. they are a copy of $\mathbb Z$, translated and dilated by $\delta$.
