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Evaluation of lattice sums via telescoping over topographs

Nikita Kalinin

TL;DR

The paper develops a unified, elementary telescoping method on Conway topographs to evaluate lattice-sum series indexed by topograph vertices. By cutting along a root edge and summing reciprocal products of adjacent region or edge labels, it yields explicit boundary expressions depending only on the root and the discriminant D, unifying Hurwitz's negative-discriminant formula, the indefinite case of Duke–Imamoḡlu–Tóth, and sums arising in Mordell–Tornheim and Hata's series. The approach provides closed forms for a broad family of sums, interprets them geometrically through arcsin/arctan/arcsinh/arctanh terms, and connects to modular graph functions in string theory, with corollaries for class numbers in square and non-square discriminants (e.g., 2\log\varepsilon_D) and a formula for h(m^{2}) when m>1. Overall, the results offer a cohesive framework linking number theory, hyperbolic/trigonometric relations, and string-theoretic lattice sums, enabling new analytic proofs and algebraic insights across discriminants.

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.

Evaluation of lattice sums via telescoping over topographs

TL;DR

The paper develops a unified, elementary telescoping method on Conway topographs to evaluate lattice-sum series indexed by topograph vertices. By cutting along a root edge and summing reciprocal products of adjacent region or edge labels, it yields explicit boundary expressions depending only on the root and the discriminant D, unifying Hurwitz's negative-discriminant formula, the indefinite case of Duke–Imamoḡlu–Tóth, and sums arising in Mordell–Tornheim and Hata's series. The approach provides closed forms for a broad family of sums, interprets them geometrically through arcsin/arctan/arcsinh/arctanh terms, and connects to modular graph functions in string theory, with corollaries for class numbers in square and non-square discriminants (e.g., 2\log\varepsilon_D) and a formula for h(m^{2}) when m>1. Overall, the results offer a cohesive framework linking number theory, hyperbolic/trigonometric relations, and string-theoretic lattice sums, enabling new analytic proofs and algebraic insights across discriminants.

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 . Our theorems are of the following spirit: we cut a topograph along an edge (called the root) into two parts, and then sum (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.

Paper Structure

This paper contains 20 sections, 22 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Formulas related to the class number
  3. String-theory formulas
  4. New results
  5. Telescoping-over-topograph method of proof
  6. Comparison with existing results
  7. Plan of the paper
  8. Topographs
  9. Useful identities in a topograph
  10. Positive-definite case, $D<0$
  11. Hurwitz series
  12. Mordell-Tornheim series
  13. Proof of Theorem \ref{['thm_5']}, first part
  14. Duality: inside and outside the circle, Theorem \ref{['thm_5']}, second part
  15. Indefinite case, $D>0$
  16. ...and 5 more sections

Key Result

Theorem 1

For $D<0$, a fundamental discriminant, all the terms in the following sum are positive and this sum converges to $h(D)$: where

Figures (4)

  • Figure 1: Illustration of $\mathcal{T}'$, an upper half of $\mathcal{T}$ relative to a root $E$, a subtree $\mathcal{T}_4"$ (in red) with a crown $C_4$ consisting of the leaves that intersect the dashed line. For each vertex $V$, we may consider the labels $r(V), s(V), t(V)$ on the adjacent regions and the labels $e(V), f(V), g(V)$ on the adjacent edges.
  • Figure 2: Left: local picture near a vertex corresponding to a superbase $\{v_1,v_2,v_3\}$. Right: local picture near an edge, with labels associated to the quadratic form $q(v)=\|v\|^2$.
  • Figure 3: Topographs locally.
  • Figure 4: Local structure of a topograph near a vertex.

Theorems & Definitions (43)

  • Theorem 1: Hurwitz,hurwitz1905darstellung
  • Example 1
  • Theorem 2: duke2021class, Theorem 3, p. 3997
  • Theorem 3: o2024topographs, Theorem 9.1
  • Theorem 4: o2024topographs, Theorem 9.2
  • Definition 1
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 1
  • ...and 33 more