On a Mertens-type conjecture for number fields (Q6659645)

For a number field \(K\) let \(\mu\) denote the Möbius function assigning an integer to each integer ideal \( \mathfrak{a}\) according to the formula \N\[\N\mu\big(\mathfrak{p}^k\big)=\begin{cases}1 &\text{ if}\ k=0,\\\N-1& \text{ if}\ k=1,\\\N0 &\text{ if}\ k\geqslant 2,\end{cases} \N\]\Nfor prime ideals \(\mathfrak{p}\). Let \N\[\NM_K(x)=\sum_{N(\mathfrak{a})\leqslant x}\mu(\mathfrak{a}) \N\]\Nbe the Mertens function.\N\NThe authors of the paper consider problems related to the behavior of the Mertens function. For instance, they prove that the naïve Mertens-type conjecture \N\[\N-1\leqslant\liminf_{x\rightarrow\infty}\frac{M_K(x)}{\sqrt{x}}\leqslant\limsup_{x\rightarrow\infty}\frac{M_K(x)}{\sqrt{x}}\leqslant 1 \N\]\Nis false for any imaginary extension of \(\mathbb{Q}\) with exception \(K=\mathbb{Q}(\sqrt{-3})\).