A note on a result of Abercrombie
The author considers the following divisor problem. Given a real number \(\theta\geq 1\), and an integer \(k\geq 2\), put \(D_k(\theta;x)=\sum_{\theta n\leq x}d_k\left (\lfloor n\theta\rfloor\right)\). It is proved in the paper that for almost all \(\theta\geq 1\) we have \[ D_k(\theta;x)=\theta^{-1}D_k(1;x)+ O(x^{(k-1)/k+\varepsilon}), \] where the \(O\)-constant depends on \(\theta\) and \(\varepsilon\). This improves and extends an earlier result by \textit{A. G. Abercrombie} [Acta Arith. 70, 195-207 (1995; Zbl 0817.11013)] who considered only the case \(k=2\) and proved the exponent \(5/7+\varepsilon\) in the exponent of the error term.
scientific article; zbMATH DE number 1091410