A two-sided omega-theorem for an asymmetric divisor problem (Q919402)

Let \(d_{a,b}(\ell,k;n)\) denote the number of solutions \((n_1,n_2)\) with \(n^a_1n^b_2=n\), \(n_2\equiv \ell \pmod k\), where \(a\ne b\), \(\ell \le k\) and \(n\) are natural numbers. Let \(\Delta_{a,b}(\ell,k;x)\) be the error term in the asymptotic representation of \(\displaystyle\sum_{n\leq x}d_{a,b}(\ell,k;n)\). In the unrestricted case \(k=\ell =1\) \textit{J. L. Hafner} [J. Number Theory 15, 36--76 (1982; Zbl 0495.10027)] has proved a very sharp \(\Omega_+\)- and \(\Omega_-\)-result. \textit{H. Menzer} and \textit{W. G. Nowak} [Manuscr. Math. 64, No. 1, 107--119 (1989; Zbl 0674.10037)] have proved a lower bound in the general case. In this paper the general case is considered under the restrictions \(0<1/k<1/6\) or \(1/2<1/k<5/6\). This is a strange case, because the author can prove a much stronger \(\Omega_{\pm}\)-result compared with Hafner's estimation.