On The Global Renormalization and Regularization of Several Complex Variable Zeta Functions by Computer

This review concerns the resolution of a special case of Knizhnik-Zamolodchikov equations ($K{Z}_3$) using our recent results on combinatorial aspects of zeta functions on several variables and software on noncommutative symbolic computations. In particular, we describe the actual solution of $(K{Z}_3)$ leading to the unique noncommutative series, ${\mathrm{\Phi }}_{KZ}$, so-called Drinfel’d associator (or Drinfel’d series). Non-trivial expressions for series with rational coefficients, satisfying the same properties with ${\mathrm{\Phi }}_{KZ}$, are also explicitly provided due to the algebraic structure and the singularity analysis of the polylogarithms and harmonic sums.