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

Abstract

We study lattice sums $\sum \frac{1}{(\|x\|\|y\|\|x+y\|)^s}$ taken over $SL_+(2,\mathbb Z)$, i.e.\ the set of pairs $(x,y)$ of primitive lattice vectors in $\mathbb Z_{\geq 0}^2$ with $\det(x, y) = 1$. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on $SL_+(2,\mathbb Z)$ that yields, in particular, $$\sum_{(x,y)\in SL_+(2,\mathbb Z)} \frac{1}{\|x\|^2\,\|y\|^2\,\|x+y\|^2}=\frac{\pi}{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+\pi^3\zeta(3)$.