Let $K$ be a number field, and let $$\zeta_{K}(s):= \sum_{0 \neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$ be the Dedekind zeta function of $K$. The quantity $s_K(x):=\sum_{n \le x} a_n$ counts ideals of $O_K$ of norm up to $x$.

$\zeta_K$ is analytic in $s\ge 1$ apart from a simple pole at $s=1$, with residue given by the class number formula. The Wiener-Ikehara theorem implies: $$s_K(x) \sim c_{K} x$$ as $x\to \infty$, where $c_K$ is given by the class number formula.

Let $E_K(x):=s_K(x) - c_K x$.

When $K=\mathbb{Q}(i)$, the problem of studying $E_K$ is known as the Gauss circle problem. Proving $E_K(x) = O(x^{1/2})$ is easy, but it is believed that $E_K(x) = O_{\varepsilon}(x^{1/4 + \varepsilon})$.

What is known about $E_K(x)$ for general number fields (conditionally and unconditionally)? What is the heuristic for that?

What distinguishes the case $K=\mathbb{Q}(i)$ from other cases? (Apart from the elementary geometric interpretation)