Multizeta Calculus (I)

TL;DR

This paper develops explicit formulas for double shuffle relations involving multiple zeta values with entries 2 or 3, advancing understanding of their algebraic structure and expressing complex relations in a computable form.

Contribution

It provides general closed formulas for key double shuffle relations involving multiple zeta values with entries 2 or 3, extending previous partial results.

Findings

Explicit formulas for double shuffle relations with zeta(1), zeta(2), zeta(3), and zeta(2,1).

Formulas for zeta(3) and zeta(2,1) are each two pages long.

Advances towards effective computation of rational coefficients in multizeta relations.

Abstract

According to Hoffman's (2,3)-conjecture, the so-called double shuffle relations should imply that every multiple zeta value should express effectively in terms of multizetas whose entries are equal to either 2 or 3, with some explicitly computable rational coefficients. In February 2011, the existence of such Q-linear combinations was established by Francis Brown in all weights. Still, a desire exists to have effective access to these coefficients. In 2008, Masanobu Kaneko, Masayuki Noro and Ken'ichi Tsurumaki showed, up to weight 20, that in fact, double shuffle relations with first member equal to zeta(1), to zeta(2), to zeta(3) or to zeta(2,1) suffice. We provide general closed formulas for such four families of double shuffle relations, with second member being an arbitrary multizeta. The longest formulas, for zeta(3) and for zeta(2,1), are 2 pages long each. This prepublication is just a first step in this direction.