Combinatorial aspects of multiple zeta values

TL;DR

This work develops a combinatorial framework for multiple zeta values (MZVs) using the shuffle algebra of their iterated-integral representations, enabling exact identities and conjectures that connect shuffle structure to MZV relations. By proving Zagier's conjecture for $\zeta(\{3,1\}^n)$ and a related dressed-with-2 cyclic sum, the authors demonstrate how shuffle-based combinatorics yields closed-form evaluations in terms of powers of $\pi$. They introduce generalized Z-structures and cyclic-sum identities, supported by extensive numerical evidence up to substantial depth, and provide a pathway for discovering further MZV identities via integer-relations methods (e.g., PSLQ) and computational tools (EZ-Face). The results bridge discrete combinatorics with continuous transcendental sums, expanding the toolkit for MZV research and its applications in physics and knot theory.

Abstract

Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle product rule allows the possibility of a combinatorial approach to them. Using this approach we prove a longstanding conjecture of Don Zagier about MZVs with certain repeated arguments. We also prove a similar cyclic sum identity. Finally, we present extensive computational evidence supporting an infinite family of conjectured MZV identities that simultaneously generalize the Zagier identity.

Figures (1)

• Figure 1: Number of linearly independent integer relations for $V_{n,M}$.

Theorems & Definitions (12)

• Lemma 1
• Lemma 2
• Lemma 3
• Definition 1
• Definition 2
• Proposition 1
• Corollary 1
• Corollary 2
• Theorem 1
• Theorem 2