Counting integral ideals in a number field

Let K   be an algebraic number field. We discuss the problem of counting the number of integral ideals below a given norm and obtain effective error estimates. The approach is elementary and follows a classical line of argument of Dedekind and Weber. The novelty here is that explicit error estimates can be obtained by fine tuning this classical argument without too much difficulty. The error estimate is sufficiently strong to give the analytic continuation of the Dedekind zeta function to the left of the line R(s)=1R(s)=1 as well as explicit bounds for the residue of the zeta function at s=1s=1.