Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

TL;DR

This paper provides elementary proofs of Zagier's and Murakami's formulas for multiple zeta and t-values, facilitating understanding of their algebraic relations and supporting conjectures about their linear combinations.

Contribution

It offers new elementary proofs of key formulas for multiple zeta and t-values, simplifying previous complex arguments and enabling further algebraic investigations.

Findings

Elementary proofs of Zagier's formula for multiple zeta values

Proof of Murakami's formula for multiple t-values

Support for Brown type results on linear combinations of zeta values

Abstract

In this paper, we give elementary proofs of Zagier's formula for multiple zeta values involving Hoffman element and its odd variant due to Murakami. Zagier's formula was a key ingredient in the proof of Hoffman's conjecture. Moreover, using the same approach, we prove Murakami's formula for multiple $t$-values. This formula is essential in proving a Brown type result which asserts that each multiple zeta value is a $\mathbb{Q}$-linear combination of multiple $t$-values of the same weight involving $2$'s and $3$'s.