New bounds and progress towards a conjecture on the summatory function of $(-2)^{\Omega(n)}$

TL;DR

This paper establishes an upper bound on the summatory function involving prime factors, advancing the understanding of a conjecture related to its growth and providing new bounds on the Mertens function.

Contribution

The paper proves that the summatory function W(x) is bounded by a linear function, specifically less than 2260x, and extends bounds to a more general function W_a(x).

Findings

Proved W(x)=O(x) with explicit bound |W(x)|<2260x

Provided new explicit bounds on the Mertens function M(x)

Extended results and conjectures to functions W_a(x) for real a>0

Abstract

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}, \end{equation*} where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. We prove $W(x)=O(x)$, and in particular, that $|W(x)|<2260x$ for all $x\geq 1$. This result provides new progress towards a conjecture of Sun, which asks whether $|W(x)|<x$ for all $x\geq 3078$. To obtain our results, we computed new explicit bounds on the Mertens function $M(x)$. These may be of independent interest. Moreover, we obtain similar results and make further conjectures that pertain to the more general function \begin{equation*} W_a(x)=\sum_{n\leq x}(-a)^{\Omega(n)} \end{equation*} for any real $a>0$.