paper should lead to further studies of MZVs with arbitrary arguments. First we can consider questions related to the renormalization procedure, such as the renormalization of MZVs with arbitrary arguments, the dependence of renormalized MZVs on the regularization and renormalization. We would also like to study the extension of the double shuffle relation to renormalized MZVs, and the possible connection to rational associators in the sense of Drinfel’d  and DMR in the sense of Racinet . Possible arithmetic properties of these renormalized MZVs, such as the Kummer type congruences, are also interesting to investigate. Some of these directions will be pursued in future works.

[**Acknowledgements.**]{} Both authors thank the Max-Planck Institute for Mathematics in Bonn for the stimulating environment where this project was started, and thank Matilde Marcolli for encouragement. We also thank Dominique Manchon and Sylvie Paycha for communications on their papers  and for their comments on our paper. The first named author thanks NSF for support, and is indebted to Kurusch Ebrahimi-Fard and Dirk Kreimer for their collaborations on QFT that inspired the renormalization approach in this paper. Thanks also go to Herbert Gangl, Robert Sczech and Jianqiang Zhao for discussions and comments.

The algebraic setup
===================

We 