For a non-negative integer $N$, let $\mathcal{ Z}_{ N}:=\sum^N_{ c = 0} \mathbb{ Q} \cdot \zeta(-c,s+c)$, 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 $\mathcal{ Z}_{ N} =\bigoplus^{ N}_{ c = 0 : \text{ even}} \mathbb{ Q} \cdot \zeta(-c,s+c)$, where $\bigoplus$ is the direct sum of vector spaces. Moreover, we give a family of relations that exhaust all $\mathbb{ Q}$-linear relations on $\mathcal{ Z}_{ N}$.
Categories: math.NT