Oscillations in weighted arithmetic sums

Abstract

We examine oscillations in a number of sums of arithmetic functions involving [Formula: see text], the total number of prime factors of [Formula: see text], and [Formula: see text], the number of distinct prime factors of [Formula: see text]. In particular, we examine oscillations in [Formula: see text] and in [Formula: see text] for [Formula: see text], and in [Formula: see text]. We show for example that each of the inequalities [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is true infinitely often, disproving some hypotheses of Sun.