On the speed of convergence to limit distributions for Dedekind zeta-functions of non-Galois number fields

Abstract

We evaluate the speed of convergence in the Bohr-Jessen type of limit theorem on the value-distribution of Dedekind zeta-functions of number fields. When $K$ is a Galois number field, the Euler product of the corresponding Dedekind zeta-function $\zeta_K(s)$ is convex, hence the evaluation can be done similarly to the case of the Riemann zeta-function. However, when $K$ is non-Galois, some new ideas (based on the Artin-Chebotarev density theorem etc) are necessary, because the corresponding $\zeta_K(s)$ is not always convex.