We examine oscillations in a number of sums of arithmetic functions involving $\Omega(n)$, 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 $S_\alpha(x) = \sum_{ n\leq x} (-1)^{ n - \Omega(n)}/n^\alpha$ and in $H_\alpha(x) = \sum_{ n\leq x} (-1)^{ \omega(n)}/n^{ \alpha}$ for $\alpha\in[0,1]$, and in $W(x)=\sum_{ n\leq x} (-2)^{ \Omega(n)}$. 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.
Categories: math.NT