Evaluation of lattice sums via telescoping over topographs

Abstract

Topographs, introduced by Conway in 1997, are infinite trivalent planar trees used to visualize the values of binary quadratic forms. In this work, we study series whose terms are indexed by the vertices of a topograph and show that they can be evaluated using telescoping sums over its edges. Our technique provides arithmetic proofs for modular graph function identities arising in string theory, yields alternative derivations of Hurwitz-style class number formulas, and provides a unified framework for well-known Mordell-Tornheim series and Hata's series for the Euler constant $γ$. Our theorems are of the following spirit: we cut a topograph along an edge (called the root) into two parts, and then sum $\frac{1}{rst}$ (the reciprocal of the product of labels on regions adjacent to a vertex) over all vertices of one part. We prove that such a sum is equal to an explicit expression depending only on the root and the discriminant of the topograph.