Sum formulas of mltiple zeta values with arguments are multiple of a positive integer

TL;DR

This paper derives formulas for sums of multiple zeta values with arguments as multiples of an integer, settling a conjecture for the case m=4 and providing explicit evaluations for small even m.

Contribution

It develops a new formula expressing these sums in terms of specific multiple zeta values, confirming Genčev's conjecture for m=4 and evaluating for small even m.

Findings

Derived formulas for E(mn,k) in terms of zeta values

Settled Genčev's conjecture for m=4

Explicit evaluations for small even m (up to 8)

Abstract

For $k\leq n$, let $E(mn,k)$ be the sum of all multiple zeta values of depth $k$ and weight $mn$ with arguments are multiples of $m\geq 2$. More precisely, $E(mn,k)=\sum_{|\boldsymbol{\alpha}|=n}\zeta(m\alpha_1,m\alpha_2,\ldots, m\alpha_k)$. In this paper, we develop a formula to express $E(mn,k)$ in terms of $\zeta(\{m\}^p)$ and $\zeta^\star(\{m\}^q)$, $0\leq p,q\leq n$. In particular, we settle Gen\v{c}ev's conjecture on the evaluation of $E(4n,k)$ and also evaluate $E(mn,k)$ explicitly for small even $m\leq 8$.