On the mean square of the error term for Dedekind zeta functions

Let $K_n$ be a number field of degree $n$ over $\mathbb Q$. Denote by $D(x,K_n)$ the number of all non-zero integral ideals in $K_n$ with norm $\leq x$. The Dedekind zeta function $\zeta_{K_n}(s)$ is a meromorphic function with a simple pole at $s=1$, with residue, say, $\Lambda_n$. Define
$$ \Delta(x, K_n)=D(x, K_n)-\Lambda_n x. $$
The history of estimates of $\Delta(x,K_n)$ begins with
$$ \Delta (x, K_n)\ll x^{1-\frac1n}\qquad\text{(Weber (1896))} $$
and
$$\Delta(x, K_n)\ll x^{\frac{n-1}{n+1}}\qquad\text{(Landau (1917))}. $$
If $n>2$, then
$$ \int^x_1\Delta(y, K_n)^2\,dy\ll x^{3-\frac4n}\log^nx, $$
which is a result of Chandrasekharan and Narasimhan (1964).
In this paper the following new results are obtained.
1) For $K_4=\mathbb Q(\root4\of{m})$, $m>1$ is square-free, the author proves
$$ x^{\frac74}\ll\int^x_1\Delta(y,K_4)^2dy\ll x^{\frac74+\varepsilon}. $$

2) For $K_6$, the normal closure of a cubic field $K_3$ with the Galois group $S_3$ and discriminant $\Delta<0$, the author proves
$$ x^{\frac{11}6}\ll\int^x_1\Delta(y,K_6)^2\,dy\ll x^{2+\varepsilon}. $$