A study on multiple zeta values from the viewpoint of zeta-functions of root systems

TL;DR

This paper explores multiple zeta values through zeta-functions linked to root systems, revealing new formulas, parity results, and sum formulas by leveraging Weyl group symmetries and root system structures.

Contribution

It introduces a novel approach connecting MZVs with root system zeta-functions, leading to new formulas and generalizations of known results.

Findings

Witten's volume formula coincides with Hoffman-Zagier formula for MZVs.

New parity results for double and triple zeta values.

Refined sum formulas for MZVs of arbitrary depth.

Abstract

We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new formulas which especially include the parity results of double and triple zeta values. As another important application, we give certain refinement of restricted sum formulas, which gives restricted sum formulas among MZVs of an arbitrary depth $r$ which were previously known only in the cases of depth $2,3,4$. Furthermore, considering a sub-root system of type $B_r$ analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas.