Problem 1122

Erdős Problem #1122: declared status 'open'. Formalized: no. Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let$$A=\{ n \geq 1: f(n+1)< f(n)\}.$$If $\lvert A\cap [1,X]\rvert =o(X)$ then must $f(n)=c\log n$ for some $c\in \mathbb{R}$? Prize: no. Tags: number theory.