the right hand side is defined in Theorem . Further, $\gzeta(\vec{s})$ agrees with $\zeta(\vec{s})$ whenever the later is defined by analytic continuation.\
    [ ${}$ ]{} Furthermore,

5.  the set $\{\gzeta(\vec{s})\big| \vec{s}\in \ZZ^k_{>0}\}$ satisfies the quasi-shuffle relation;

6.  the set $\{\gzeta(\vec{s})\big| \vec{s}\in \ZZ^k_{\leq 0}\}$ satisfies the quasi-shuffle relation.

The items of this theorem will be proved in the rest of this paper.

() is a restatement of Theorem . () is Theorem  combined with Proposition . () and the first statement of () are contained in Corollary . The second statement of () is Proposition . () is just Corollary . () is just Theorem .

Renormalized multiple zeta values with positive arguments
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We first take care of the easy case when MZVs are define by the convergence of the nested sums.

Let $\vec{s}=(s_1,\cdots,s_k)$ with positive integers $s_1,\cdots,s_k$ and $s_1>1$. We have $\zeta(\wvec{\vec{s}}{\vec{r}})=\zeta(\vec{s})$, independent of the choice of $\vec{r}\in \ZZ_{>0}^k$. In particular, $\gzeta(\vec{s})=\zeta(\vec{s})$.

For such an $\vec{s}$, $Z(\wvec{\vec s}{\vec r};\vep)$ is uniformly convergent in $(-\infty, 0]$, and the summands are continuous functions. So $Z(\wvec{\vec s}{\vec r};\vep)$ is continuous in $(-\infty, 0]$. Therefore, the Laurent series of $Z(\wvec{\vec s}{\vec r};\vep)$ is a power series and, by Theorem , $\zeta(\wvec{\vec{s}}{\vec{r}})=\lim_{\vep\to 0} \uni{Z}_+(\wvec{\vec{s}}{\vec{r}};\vep)
