Evaluating lattice sums via telescoping on $S{L}_{+}(2,\mathbb{Z})$: a short proof of $\sum \frac{1}{|x{|}^2|y{|}^2|x+y{|}^2}=\frac{\pi }{4}$ and Zagier's identity

Nikita Kalinin

We study lattice sums $\sum \frac{1}{(\Vert x\Vert \Vert y\Vert \Vert x+y\Vert {)}^s}$ taken over $S{L}_{+}(2,\mathbb{Z})$, i.e.\ the set of pairs $(x,y)$ of primitive lattice vectors in ${\mathbb{Z}}_{\ge 0}^2$ w…