On a basis for Euler-Zagier double zeta functions with non-positive components

Abstract

For a non-negative integer $N$, let ${\mathcal{Z}}_N:={\sum }_{c=0}^N\mathbb{Q}\cdot \zeta (-c,s+c)$, where the right-hand side is the vector space spanned by the Euler-Zagier double zeta functions over $\mathbb{Q}$. In this paper, we show that ${\mathcal{Z}}_N={⨁}_{c=0:\text{even}}^N\mathbb{Q}\cdot \zeta (-c,s+c)$, where $⨁$ is the direct sum of vector spaces. Moreover, we give a family of relations that exhaust all $\mathbb{Q}$-linear relations on ${\mathcal{Z}}_N$.

13 pages