Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity
Nikita Kalinin
Published: 2024/10/10
Abstract
We study lattice sums $\sum 1/(|x||y||x+y|)^s$ taken over $SL_+(2,\ZZ)$, i.e. the set of pairs $(x,y)$ of primitive lattice vectors in $\ZZ_{\geq 0}^2$ with $\det(x, y) = 1$. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on $SL_+(2,\ZZ)$ that yields, in particular, $$\sum_{(x,y)\in SL_+(2,\ZZ)} \frac{1}{|x|^2|y|^2|x+y|^2}=\frac{\pi}{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+\pi^3\zeta(3)$.