Im trying to understuand the proof of this Lemma
Let $\Omega$ be a lattice in $\Bbb C.$ The series :$$L:=\underset{0\neq\rho\in\Omega}{\sum}\frac{1}{|\rho|^t}$$ is absolutely convergent for $t>2.$
In the proof I find this statement ; "We note that the number of elements of $\Omega$ such that $|\rho|$ is between the consecutive integers $n$ and $n + 1$ is $O(n)$ so the convergence of $L$ is reduced to the convergence of $\sum_{r>0}\frac{1}{r^{t-1}}$ where $r$ is a positive integer, and we know that this converges for $t − 1 > 1$"
Can someone explain to me : why $\sharp\{\rho\in\Omega:n\leq|\rho|\leq n+1\}=O(n)$ ? and how I can use this to conclude that $L$ converge ?
thanks!
Suppose for a moment that $\Omega$ is the integer lattice. Then the region described is an annulus centered at the origin, of inner radius $n$ and outer radius $n + 1$. The area of the outer disk is $\pi (n+1)^2$; the number of "cells" touching the outer disk is at most $\pi (n+2)^2$. Similarly, the number of cells completely within the inner disk is at least $\pi (n-2)^2$. So the number of cells meeting the annulus is no more that $\pi ( (n+2)^2 - (n-2)^2)$, which is $O(n)$.
(a "cell" is a unit square centered at a lattice point)
For your case, it's pretty much the same argument...but with a slightly different metric.
I hope this helps get you on your way, but if not, please try to explain where you got stuck.