On certain sums of number theory

TL;DR

This paper investigates sums involving number-theoretic functions, improving bounds for certain cases and establishing new results for higher divisor functions, advancing understanding of these sums in analytic number theory.

Contribution

The paper provides improved estimates for sums involving the von Mangoldt and divisor functions, and introduces new bounds for higher divisor functions, surpassing previous barriers.

Findings

Enhanced bounds for sums with von Mangoldt function

New results for divisor functions with r ≥ 3

Breakthrough in the 1/2 barrier for these sums

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.