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

Abstract

For $k\le n$, let $E(mn,k)$ be the sum of all multiple zeta values of depth $k$ and weight $mn$ with arguments are multiples of $m\ge 2$. More precisely, $E(mn,k)={\sum }_{|\mathbf{\alpha }|=n}\zeta (m{\alpha }_1,m{\alpha }_2,\dots ,m{\alpha }_k)$. In this paper, we develop a formula to express $E(mn,k)$ in terms of $\zeta (\{m{\}}^p)$ and ${\zeta }^{\star }(\{m{\}}^q)$, $0\le p,q\le n$. In particular, we settle Genčev's conjecture on the evaluation of $E(4n,k)$ and also evaluate $E(mn,k)$ explicitly for small even $m\le 8$.

16 pages