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

Key Points

• The aim is to evaluate lattice sums and provide a concise proof of Zagier’s identity using a telescoping method.
• Introduced a new telescoping method for evaluating sums
• Studied pairs of primitive lattice vectors in SL+(2, ℤ)
• Analyzed convergence of determinant weighted sums
• Confirmed convergence for the evaluated lattice sums
• Derived the identity ∑ 1/∥x∥²∥y∥²∥x + y∥² = π/4
• Presented a short proof of Zagier’s identity