Oscillations in weighted arithmetic sums

TL;DR

This paper investigates oscillations in sums involving the number of prime factors of integers, demonstrating that certain inequalities hold infinitely often, thereby disproving some of Sun's hypotheses.

Contribution

It introduces new results on the oscillatory behavior of arithmetic sums involving prime factor counts, challenging existing conjectures.

Findings

Certain inequalities hold infinitely often, such as $S_0(x)<0$ and $S_0(x)>3.3\sqrt{x}$.

Oscillations are shown in sums involving $inom{ ext{prime factors}}{}$, disproving some prior hypotheses.

The paper provides new insights into the distribution of arithmetic functions related to prime factors.

Abstract

We examine oscillations in a number of sums of arithmetic functions involving $\Omega(n)$, the total number of prime factors of $n$, and $\omega(n)$, the number of distinct prime factors of $n$. In particular, we examine oscillations in $S_\alpha(x) = \sum_{n\leq x} (-1)^{n - \Omega(n)}/n^\alpha$ and in $H_\alpha(x) = \sum_{n\leq x} (-1)^{\omega(n)}/n^{\alpha}$ for $\alpha\in[0,1]$, and in $W(x)=\sum_{n\leq x} (-2)^{\Omega(n)}$. We show for example that each of the inequalities $S_0(x)<0$, $S_0(x)>3.3\sqrt{x}$, $S_1(x)>0$, and $S_1(x)\sqrt{x}<-3.3$ is true infinitely often, disproving some hypotheses of Sun.