For integral ideals \(\mathfrak a\) of a quadratic number field \(K\) over \(\mathbb Q\), define the Liouville function by \(\lambda(\mathfrak a)=(-1)^{e_1+\ldots+e_r}\), if \(\mathfrak a=\mathfrak p_1^{e_1}\dots\mathfrak p_r^{e_r}\) is the canonical factorization, and put \(\ell (\mathfrak a)=1\) if \(\mathfrak p^2\mid \mathfrak a\) for every prime ideal \(\mathfrak p\) which divides \(\mathfrak a\), \(\ell (\mathfrak a)=0\) otherwise. Motivated by the simple generating function
\[ \sum_{\mathfrak a}\lambda (\mathfrak a)\ell (\mathfrak a)N(\mathfrak a)^{-s}=\zeta_K(2s)(\zeta_K(3s))^{-1}\quad (\text{Re}\, s>1), \]
one can ask for an asymptotic formula for the sum of coefficients
\[ S(x)=\sum_{N(\mathfrak a)\leq x}\lambda (\mathfrak a) \ell (\mathfrak a). \]
In a recent paper, U. B. Zhanbyrbaeva and N. A. Fugelo [Izv. Akad. Nauk Kaz. SSR, Ser. Fiz. Mat. 1982, No. 1, 60–64 (1982; Zbl 0487.10038)] proved for the special case \(K=\mathbb Q(i)\) that
\[ S(x)=\pi \zeta_K(3/2)^{-1} \sqrt{x}+O\left(\root 3\of {x} \exp(-c'(\log x)^\alpha)\right) \] for some \(c'>0\) and any \(\alpha\), using zero density results on the Dedekind zeta-function \(\zeta_K\) involved.
In the present little note it is pointed out that a direct convolution argument leads from known results on the distribution of ideals and on the Möbius function in \(K\) to the refinement
\[ S(x)=\rho \zeta_K(3/2)^{-1} \sqrt{x}+O(\root 3\of {x} \exp \left(-c(\log x)^{3/5} (\log \log x)^{-1/5}\right)\quad (c>0) \]
where \(\rho\) is the residue of \(\zeta_K\)(s) at \(s=1\).
(Remark: Unfortunately there are several seriously disturbing printing errors, even in the statement of the above result.)