On certain sums of number theory

Olivier Bordellès

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arXiv · 2009.05751

On certain sums of number theory

Olivier Bordellès

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Abstract

We study sums of the shape $\sum_{n \leqslant x} f \left( \lfloor x/n \rfloor \right)$ 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 $r \geqslant 3$, breaking the $\frac{1}{2}$-barrier in each case. The functions $f=\mu^2$, $f=2^\omega$ and $f=\omega$ are also investigated.

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Olivier Bordellès. 2020-09-12. On certain sums of number theory. https://arxiv.org/abs/2009.05751

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