Evaluating lattice sums via telescoping on SL+(2, ℤ): A short proof of ∑ 1 ∥x∥2∥y∥2∥x+y∥2 = π 4 and of Zagier’s identity

We study lattice sums [Formula: see text] taken over [Formula: see text], i.e. the set of pairs [Formula: see text] of primitive lattice vectors in [Formula: see text] with [Formula: see text]. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on [Formula: see text] that yields, in particular, [Formula: see text] and a short proof of Zagier’s identity [Formula: see text].