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

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 $a>0$.