Oscillations in weighted arithmetic sums

We examine oscillations in a number of sums of arithmetic functions involving $\mathrm{\Omega }(n)$, 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 $S_{\alpha }(x)={\sum }_{n\le x}(-1{)}^{n-\mathrm{\Omega }(n)}\mathrm{/}{n}^{\alpha }$ and in $H_{\alpha }(x)={\sum }_{n\le x}(-1{)}^{\omega (n)}\mathrm{/}{n}^{\alpha }$ for $\alpha \in [0,1]$, and in $W(x)={\sum }_{n\le x}(-2{)}^{\mathrm{\Omega }(n)}$. 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.