Lattice sums

Lattice sums

January 15, 2023

In this entry we will evaluate symmetric infinite sums of the type

$\begin{align*} S=\sum_{(n,m) \neq (0,0)}\frac{1}{(n^2+m^2)^s} \end{align*}$

To this end we will rely on the previous established result (proved here)

$\begin{align*} \left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2=2 \sum_{n=-\infty}^\infty \frac{q^n}{1+q^{2n}} \end{align*}$

Click here for the proof.

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