On Zagier's Conjecture About the Inverse of a Matrix Related to Double Zeta Values

TL;DR

This paper proves Zagier's conjecture on the inverse of a specific matrix related to double zeta values, enabling new expressions of products of single zeta values in terms of double zeta values using elementary methods.

Contribution

It provides a proof of Zagier's conjecture on the inverse matrix, offering a new formula to relate products of single zeta values to double zeta values.

Findings

Explicit formula for the inverse matrix A_K

Expression of products of zeta(2r) and zeta(2K+1-2r) in terms of double zeta values

Elementary proof technique for Zagier's conjecture

Abstract

We prove a conjecture of Zagier about the inverse of a $(K-1)\times (K-1)$ matrix $A=A_{K}$ using elementary methods. This formula allows one to express the the product of single zeta values $\zeta(2r)\zeta(2K+1-2r)$, $1\leq r\leq K-1$, in terms of the double zeta values $\zeta(2r,2K+1-2r)$, $1\leq r\leq K-1$ and $\zeta(2K+1)$.