the motivic viewpoint of Drinfeld associators, and those by Deligne and Goncharov [@DG] who study the motivic fundamental groups of ${{\mathbb P}}^1-(\{0,\infty\}\cup\mmu_N)$ by using the theory of mixed Tate motives over $S$-integers of number fields, where $\mmu_N$ is the group of $N$th root of unity.

Fix an $N$th root of unity $\mu=\mu_N:=\exp(2\pi
\sqrt{-1}/N)$. The *level* $N$ MPVs are defined by $$\label{equ:z}
L_N(s_1,\dots,s_\ell|i_1,\dots,i_\ell):=
Li_{s_1,\dots,s_\ell}(\mu^{i_1},\dots,\mu^{i_\ell}).$$ We will always identify $(i_1,\dots,i_\ell)$ with $(i_1,\dots,i_\ell)
\pmod{N}.$ It is easy to see from that a MPV converges if and only if $(s_1,\mu^{i_1})\ne (1,1).$ Clearly, all level $N$ MPVs are automatically of level $Nk$ for any positive integer $k$. For example when $i_1=\cdots=i_\ell=0$ or $N=1$ we get the multiple zeta values $\zeta(s_1,\dots,s_\ell)$. When $N=2$ we recover the alternating Euler sums studied in [@BBB2; @Zesum]. To save space, if a substring $S$ repeats $n$ times in the list then $\{S\}^n$ will be used. For example, $L_N(\{2\}^2|\{0\}^2)=\zeta(2,2)=\pi^4/120$.

Standard conjectures in arithmetic geometry imply that $\Q$-linear relations among MVPs can only exist between those of the same weight. Let $\MPV(w,N)$ be the $\Q$-span of all the MPVs of weight $w$ and level $N$ whose dimension is denoted by $d(w,N)$. In general, to determine $d(w,N)$ precisely is a very difficult problem 