were obtained in the these papers. But they fell short of the analogous properties of the usual MZVs, especially the double shuffle relations.

In this paper, we adapt a renormalization procedure (dimensional regularization plus minimal subtraction) in quantum field theory (QFT) to define the values of multiple zeta functions $\zeta(s_1,\cdots,s_k)$ at $(s_1,\cdots,s_k)$ when $s_i$, $1\leq i\leq k$, are all non-positive or all positive, that we expect to further extend to when $s_i$ are [*arbitrary integers*]{}. For our purpose, the dimensional regularization of Feynman integrals is replaced by a regularization (or deformation) of infinite series that has occurred in the study of Todd classes for toric varieties .

The renormalization procedure of QFT was put in the framework of Hopf algebra and Rota-Baxter algebra by the recent works of Connes and Kreimer , continued in , and thus made possible for applications beyond QFT. A fundamental result in this framework is the Algebraic Birkhoff Decomposition (Theorem ). It states that for a given triple $(\calh, R, \phi)$ consisting of

-   a connected filtered Hopf algebra $\calh$,

-   a commutative Rota-Baxter algebra $R$ on which the Rota-Baxter operator $P:R\to R$ is idempotent, and

-   an algebra homomorphism $\phi: \calh\to R$,

there are 