On the proximity of multiplicative functions to the function counting prime factors with multiplicity

Abstract

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\#\{n\le x:f(n)=g(n)\}$. We establish a lower bound for $E(\Omega ,g,x)$.