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

Abstract

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 ${\mathbb{Z}}_{\ge 0}^2$ with $\det (x,y)=1$. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on $S{L}_{+}(2,\mathbb{Z})$ that yields, in particular, $$\sum _{(x,y)\in S{L}_{+}(2,\mathbb{Z})}\frac{1}{\Vert x{\Vert }^2\Vert y{\Vert }^2\Vert x+y{\Vert }^2}=\frac{\pi }{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+{\pi }^3\zeta (3)$.

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