So, I have a square integer lattice, that is $p\in 2\pi\mathbb Z^2\setminus\{0\}$, and some positive real number $z\in\mathbb R^{>0}$ and the series $$ \sum_{p\in 2\pi\mathbb Z^2\setminus\{0\},|p|\leq|z|^{-1}} \frac 1 {|p|^2}. $$ This means the summation includes all lattice points $p$ that are inside the circle with radius $\frac 1 {|z|}.$
For my bigger proof to work, I need this series to converge uniformly (in regards to $z$ in a small neighborhood of $0$). Unfortunately, I don't really know how to tacle this problem. The only thoughts that I have had so far, have led me to believe that this series does not converge.