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

TL;DR

The paper studies lattice sums over $SL_+(2,\mathbb Z)$, focusing on $S=\sum_{(x,y)\in SL_+(2,\mathbb Z)} \frac{1}{\|x\|\|y\|\|x+y\|}$ and related determinant-weighted variants. It introduces a novel telescoping method on the Stern–Brocot tree to achieve cancellations, yielding a precise evaluation of key sums and enabling a concise proof of Zagier's lattice sum identity. It establishes convergence of the basic and determinant-weighted sums, derives an explicit constant $\frac{\pi}{4}$ for the squared-norm case, and presents a determinant-weighted generalization $S^{\det}(s,z)=\frac{6\pi}{y^3}\zeta(s+3)\zeta(s+2)$ that extends to arbitrary lattices. The work suggests a broad, potentially automorphic-analytic framework for telescoping sums on lattice bases with applications to modular graph functions and Eisenstein-series identities.

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, $$\sum_{(x,y)\in SL_+(2,\mathbb Z)} \frac{1}{\|x\|^2\,\|y\|^2\,\|x+y\|^2}=\fracπ{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+π^3ζ(3)$.

Theorems & Definitions (24)

• Conjecture 1
• Theorem 1
• proof
• Remark 1
• Remark 2
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4