&\\
& \to \mbox{renormalized directional MZVs}\ \zeta(\wvec{\vec{s}}{\vec{r}}) \to \mbox{renormalized MZVs}\ \gzeta(\vec{s}) &\end{aligned}$$ The concepts of regularization and renormalization have already been introduced to the study of MZVs by Ihara-Kaneko-Zagier  to take care of the divergency of the MZVs $\zeta(s_1,\cdots,s_k)$ with $s_1=1$. As a part of their process, the natural algebra homomorphism from the the quasi-shuffle algebra for convergent MZVs to the algebra of convergent MZVs is extended to an algebra homomorphism from a larger quasi-shuffle algebra to an extension of the algebra of convergent MZVs. Thus they obtained their extended MZVs as an [**algebraic continuation**]{} (we thank Robert Sczech for suggesting this term) in the sense that their extended MZVs preserves the quasi-shuffle relation. From this point of view, we obtain our renormalized MZVs as an algebraic continuation that goes beyond theirs to cover the MZVs with all non-positive arguments. In a weak sense it covers arbitrary arguments (Definition ). More recently, Manchon and Paycha  have considered renormalization of MZVs from the point of view of Chen integrals and Chen sums of symbols using a similar renormalization approach in the spirit of Connes and Kreimer. The two approaches should be related though the exact link is still not clear.

This 