On the Dedekind zeta function

Let $K_n$ be a number field of degree $n$ over $\mathbb Q$. Denote by $A_{K_n}(x)$ the number of ideal with norm $\leq x$. Landau's classical estimate is
$$ A_{K_n}(x)=\Lambda_nx+O(x^{(n-1)/(n+1)}). $$
In this paper the error term is improved for the non-normal field $K_4=\mathbb Q(\root4\of m)$ and for $K_6$, the normal closure of a cubic field $K_3$ with the Galois group $S_3$.