Let \[ t_2(n) = \sum_{|{\mathcal G}|=n,r({\mathcal G})\leq2}\tau({\mathcal G}), \] where \(\tau({\mathcal G})\) denotes the number of subgroups of a finite abelian group \({\mathcal G}, r({\mathcal G})\) is the rank of \({\mathcal G}\), and \(|{\mathcal G}|\) is the order of \({\mathcal G}\). The group \({\mathcal G}\) has rank \(r\) if \({\mathcal G}\cong \mathbb Z/n_1\mathbb Z \otimes \cdots \otimes \mathbb Z/n_r\mathbb Z,\) where \(n_j\mid n_{j+1}\) for \(j = 1,\cdots,r - 1\). We set \[ T(x) =\sum_{n\leq x}t_2(n) = \sum_{|{\mathcal G}|\leq x,r({\mathcal G})\leq2}\tau({\mathcal G}) \] so that one has \(T(x) = K_1x\log^2x + K_2x\log x + K_3x + \Delta(x)\), where \(K_j\) are effective constants and \(\Delta(x)\) is to be considered as the error term in the asymptotic formula for \(T(x)\).
In the present work the authors obtain the bound \(\Delta(x) \ll x^{31/43+\varepsilon}\). This is obtained by using convolution (the Dirichlet hyperbola method) and the sharpest known asymptotic formulas for the summatory functions of \(\sigma(n) = \sum_{d|n}d\) and \(d^*(n)=\sum_{n_1n_2n_3^2n_4^2=n}1\), since the relevant generating Dirichlet series in this problem turns out to be \[ \zeta^2(s)\zeta^2(2s)\zeta(2s-1)\prod_p(1+p^{-2s}-2p^{-3s}). \] In the meantime H. Menzer [Proc. Conf. Anal. Number Theory, Univ. Wien, 181-188 (1996; Zbl 0879.11049)] improved the above bound to \( \Delta(x) \ll x^{9/14+\varepsilon}, \) by using two estimates in the three-dimensional asymmetric divisor problem. G. Bhowmik and J. Wu [Arch. Math. 69, 95-104 (1997)] used another method, based on a weighted three-dimensional divisor problem to obtain a further improvement, namely \(\Delta(x) \ll x^{5/8}\log^4x\) (note that 31/43 = 0.720930\(\dots\), 9/14 = 0.642857\(\dots\), 5/8 = 0.625).