Generalized divisor problem (Q1346374)

Previous authors have studied generalizations of the divisor problem and have used a variety of methods. In particular, \textit{M. Vogts} [Math. Nachr. 124, 103-121 (1985; Zbl 0595.10034)] investigated sums of the type \((*)\) \(\sum m_ 1^{a_ 1} \dots m_ n^{a_ n}\) subject to the summation condition \(m_ 1^{b_ 1} \dots m_ n^{b_ n} \leq x\) and used elementary methods to estimate the error terms. The present authors apply analytical methods to evaluate the sum in \((*)\) when the \(a_ j\) and \(b_ j\) satisfy certain conditions too complicated to quote here, and they obtain the expected main terms plus an \(O\)-error term in the form of a smaller power of \(x\).