Explicit estimates of the weighted sum $S(x)=\sum_{n \leq x} (-2)^{\Omega(n)} \log\bigl(\frac{x}{n}\bigr).$

We study the oscillatory arithmetic function $(-2)^{\Omega(n)}$, where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. Sun conjectured a bound on its partial sums $W(x) = \sum_{n \leq x}^{} (-2)^{\Omega(n)}$ as $|W(x)| < x$ for all $x \geq 3078$. In this direction, we obtain new bounds for its logarithmically weighted average \begin{equation*}
S(x)=\sum_{n \leq x} (-2)^{\Omega(n)} \log\biggl(\frac{x}{n}\biggr). \end{equation*} Using complex-analytic methods such as the log-weighted Perron's formula, we computed the bound $|S(x)| \leq 1.6x$