Oscillations in weighted arithmetic sums

Abstract

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 $S_{\alpha }(x)={\sum }_{n\le x}(-1{)}^{n-\Omega (n)}/n^{\alpha }$ and in $H_{\alpha }(x)={\sum }_{n\le x}(-1{)}^{\omega (n)}/n^{\alpha }$ for $\alpha \in [0,1]$, and in $W(x)={\sum }_{n\le x}(-2{)}^{\Omega (n)}$. 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.

To appear in Int. J. Number Theory