Smooth L² distances and zeros of approximations of Dedekind zeta functions

We consider a family of approximations of the Dedekind zeta function ζ_K(s) of a number field K/ Q. Weighted L²-norms of the difference of two such approximations of ζ_K(s) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of ζ_K(s) can be obtained from such weighted L²-norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function.