Dyson Schwinger Equations: From Hopf algebras to Number Theory

We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive non-perturbative results for the short-distance singular sector of a renormalizable quantum field theory. We focus on the short-distance behaviour and thus discuss renormalized Green functions $G_R(\alpha,L)$ which depend on a single scale $L=\ln q^2/\mu^2$.