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The serie of function

$$\sum_{n\in\mathbb Z}\frac{1}{{(z-n)}^2}$$

converges normally in $\mathbb C\setminus\mathbb Z$ and it defines a meromorphic simply periodic function. Now let be $\Lambda$ a lattice, then the serie $$\sum_{\omega\in\Lambda}\frac{1}{{(z-\omega)}^2}$$ doesn't converge normally in $\mathbb C\setminus \Lambda$ so it doesn't define an elliptic function. Geometrically, Why does this occur? If there is a discrete line of poles is ensured the convergence but if poles form a lattice the serie doesn't converge. I have just proved the above statements with analysis tools, but i don't understand why such two series have a so different behavior.

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up vote 3 down vote accepted

Take, for example, $\Lambda=\{(n,m):n,m \in \mathbb{Z}\}$ and look at the sum for $z=0$ -- ignore $\omega=(0,0)$ for the sake of this discussion. Then you can rearrange terms so that you sum over the terms which are from the squares centered around $0$. The inner square has $8$ points contributing to the sum the next one already $16$, in general the $n$- th will contribute $2*(2n+1)+ 2*(2n-1)$ (draw a picture). So each of these squares contributes $Cn$ points, but the order of the terms is roughly $1/n^2$. In other words, the sum behaves like a harmonic series. This is basically because the number of terms of a certain magnitude is increasing, due to 2 dimensions, linearly. In one dimension the number of terms of a certain magnitude is constant.

This behaviour is similar to the behaviour you observe if you try to integrate $$\int_{|x|>1}\left(\frac{1}{|x|}\right)^k$$ in spaces of increasing dimensions. The bound for $k$, for which this converges, depends in a similar manner (for the same reason) on the dimension $n$. Technically this corresponds to a factor of $r^{n-1}$ in the volume element if written down in polar coordinates, geometrically this of course just the way the surface of the $(n-1)$-dimensional sphere scales.

In other words, this is not a question about complex analytic function. It becomes a question about these if you add terms forcing normal convergence and arrive at the Weierstrass function. But then other effects become predominant (topological ones, elliptic functions are functions on a torus).

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