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

We study lattice sums $\sum \frac{1}{(\mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }\mathrm{\parallel }x+y\mathrm{\parallel }{)}^s}$ 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, $$\sum _{(x,y)\in S{L}_{+}(2,\mathbb{Z})}\frac{1}{\mathrm{\parallel }x{\mathrm{\parallel }}^2\mathrm{\parallel }y{\mathrm{\parallel }}^2\mathrm{\parallel }x+y{\mathrm{\parallel }}^2}=\frac{\pi }{4},$$ and a short proof of Zagier’s identity $D_{1,1,1}=2E(z,3)+{\pi }^3\zeta (3)$.