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For $\vec{s}\in \ZZ_{> 0}^k\cup \ZZ_{\leq 0}^k$, define the [**renormalized MZV**]{} at $\vec{s}$ to be $$\gzeta\lp \vec{s}\rp
= \lim_{\delta \to 0^+} \zeta\lp \wvec{\vec{s}}{|\vec{s}|+\delta}\rp ,
\mlabel{eq:gmzv}$$ where, for $\vec s=(s_1,\cdots,s_k)$ and $\delta\in \RR_{>0}$, we denote $|\vec{s}|=(|s_1|,\cdots,|s_k|)$ and $|\vec s|+\delta=(|s_1|+\delta,\cdots,|s_k|+\delta).$

[Theorem  below is our main theorem. It shows that our renormalized MZVs is well-defined and is compatible with known MZVs defined by either convergence, analytic continuation or the Ihara-Kaneko-Zagier regularization. It also proves that it satisfies the quasi-shuffle relation. We are optimistic that this is in fact the only definition of $\gzeta(\vec{s})$ from $\zeta(\wvec{\vec s}{\vec r})$ with these properties and will elaborate on it in a subsequent work. ]{}

The limit in Eq. () exists for any $\vec{s}=(s_1,\cdots,s_k)\in \ZZ_{> 0}^k\cup \ZZ_{\leq 0}^k$. More precisely,

1.  when $s_i$ are all positive with $s_1>1$, we have $\zeta\lp \wvec{\vec{s}}{\vec{r}}\rp =\zeta(\vec{s})$ independent of $\vec{r}\in \ZZ_{>0}^k$. In particular, $\gzeta(\vec{s})=\zeta(\vec{s})$;

2.  when $s_i$ are all positive, we have $\gzeta(\vec{s})=\zeta\lp \wvec{\vec{s}}{\vec{s}}\rp $. Further, $\gzeta(\vec{s})$ agrees with the regularized MZV $Z^*_{\vec{s}}(T)$ defined by Ihara-Kaneko-Zagier ;

3.  when $s_i$ are all negative, we have $\gzeta(\vec{s})=\zeta\lp \wvec{\vec{s}}{-\vec{s}}\rp =\disp{\lim_{\vec{r}\to -\vec{s}} \zeta\lp \wvec{\vec{s}}{\vec{r}}\rp} ;$

4.  when $s_i$ are all non-positive, we have $\gzeta(\vec{s})= \disp{\lim _{\vec r\to -\vec s}\zeta \lp \wvec{\vec s}{\vec
    r}\rp ^{(\Sigma)}}$ where 