On a sum involving the divisor function (Q2052784)

The authors use some newer techniques to show that the following estimate is valid for the divisor function \(d\): \[\sum_{n \leq x} d\left(\left[\frac{x}{n}\right]\right)=x \sum_{m \geq 1} \frac{d(m)}{m(m+1)}+O_{\varepsilon}\left(x^{11 / 23+\varepsilon}\right)\quad (x \rightarrow \infty).\] A part of the results of the paper that is needed in the course of the proof and is interesting in itself, gives estimates for \[\mathfrak{S}_{\delta}(x, D):=\sum_{D<k \leq 2 D} d(k) \psi\left(\frac{x}{k+\delta}\right)\quad(x \geq 2,\, 1 \leq D \leq x),\] where \(\psi(t):=t-[t]-\frac{1}{2}\), and \(\delta \in\{0,1\}\).