On certain sums of number theory

Abstract

We study sums of the shape ${\sum }_{n⩽x}f\left(\lfloor x/n\rfloor \right)$ where $f$ is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when $f=\Lambda$ and $f=\tau$, and provide new results when $f={\tau }_r$ with $r⩾3$, breaking the $\frac{1}{2}$-barrier in each case. The functions $f={\mu }^2$, $f=2^{\omega }$ and $f=\omega$ are also investigated.

16 pages. In this version, the function omega has been added. Comments are welcome