Let \(\mathbb Z[i]\) denote the ring of Gaussian integers with the associated Liouville function \(\lambda(\alpha) = (-1)^{a_1 +\dots+ a_k}\) if the canonical decomposition of \(\alpha\in\mathbb Z[i]\) is \(\alpha = \rho_1^{a_1} \cdots \rho_k^{a_k}\), and let \(N(\alpha)\) denote the norm of \(\alpha\). A Gaussian integer \(\alpha\) is called square-full if \(a_1\geq 2, \dots, a_k\geq 2\). If \(\ell(\alpha)\) equals 1 when \(\alpha\) is squarefull and zero otherwise, then the authors sketch a proof of
\[ \sum_{0<N(\alpha)\leq x} \ell(\alpha) \lambda(\alpha) = Ax^{1/2} + O\left(x^{1/3}\exp(-C \log^{1/2-\varepsilon}x)\right) \tag{1} \]
with some suitable \(C > 0\), where \(A= (4\zeta(3/2)L(3/2,\chi_4))^{-1}> 0\) and \(\chi_4\) is the non-principal character mod 4. The method employed is based on the standard Perron truncated formula for sums of coefficients of Dirichlet series. The generating series in this problem turns out to be \(4\zeta(2s)L(3s,\chi_4)(\zeta(3s)L(3s,\chi_4))^{-1}\), and to obtain the error term in (1) the authors make use of a technique introduced by Y. Motohashi [Proc. Jap. Acad. 52, 477–479 (1976; Zbl 0372.10033)] and several zero-density estimates for the Riemann zeta-function. A result related to (1) is also stated.
(Reviewer’s remark: there are several misprints in the references and a factor \((\log T)^9\) in (8) missing.)