Explicit estimates of the weighted sum $S(x)={\sum }_{n\le x}(-2{)}^{\Omega (n)}\log (\frac{x}{n}).$

Abstract

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 $W(x)={\sum }_{n\le x}^{}(-2{)}^{\Omega (n)}$ as $|W(x)|<x$ for all $x\ge 3078$. In this direction, we obtain new bounds for its logarithmically weighted average \begin{equation*} S(x)=\sum_{n \leq x} (-2)^{Ω(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 $|S(x)|\le 1.6x$