our definition of $\gzeta(\vec{s})$ for $s_i \geq 1,
1\leq i\leq k$, also satisfies $\gzeta(1)=T$ (by Theorem ) and the quasi-shuffle relation (by Corollary ), the later $\gzeta(\vec{s})$ must agree with the regularized MZVs $Z^\msh_{\vec{s}}(T)$.

Renormalized multiple zeta values with non-positive arguments
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We study $\gzeta(\vec{s})$ when $\vec{s}\in \ZZ_{\leq 0}^k$. We then show that these values agrees with the special values of the multiple zeta functions with negative arguments defined by analytic continuation.

### The case of $\vec{s}=(0,\cdots,0)$

Let $\vec{s}=(s_1,\cdots,s_k)\in \ZZ_{\le 0}^k$ and $\vec{r}=(r_1,\cdots,r_k)\in \ZZ_{>0}^k$.

1.  For $k\ge 2$, we have

2.  If $\ell<k$ and $s_1=\cdots=s_\ell=0$, then $$Z\lp\wvec{\vec s}{\vec r}\rp= Z\lp \wvec{0,\cdots,0}{r_1,\cdots,r_\ell}\rp
        Z\lp \wvec{s_{\ell+1},s_{\ell+2},\cdots,s_k}{r_1+\cdots+r_{\ell+1},r_{\ell+2},\cdots,r_k}\rp.$$

3.  Each coefficient in the Laurent series expansion of $Z (\wvec{\vec s}{\vec r};\vep )$ is a rational function of the form $P(\vec r)/Q(\vec r)$, where $P$, $Q$ are in $\CC[r_1,\cdots,r_k]$ with no common factors, and are of the form $\Pi_{1\le j \leq k} (r_1+r_{2}+\cdots +r_j)^{c_{j}}$, $c_{j}\in
    \ZZ _{\ge 0}$.

() Since $s_1\leq 0$, we have This gives the proposition when $k=2$. In general, applying the induction hypothesis to the second $Z$-factor completes the proof.

() Applying the first equation of item () repeatedly, we have $$\begin{aligned}
\lefteqn{ Z(\wvec{\vec s}{\vec r};\vep) = Z(\wvec{0}{r_1};\vep )
Z(\wvec{s_2, s_3, \cdots s_{k}}{r_1+r_2, 