What question did this study set out to answer?

The aim is to evaluate lattice sums and provide a concise proof of Zagier’s identity using a telescoping method.

April 23, 2026International Journal of Number Theory0 citations

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

NKNikita KalininTechnion – Israel Institute of Technology

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

Abstract

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.

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