Erdős Problem #1122

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 \(c\in \mathbb{R}\)? The condition \(\lvert A\cap [1,X]\rvert =o(X)\) says that \(A\) has natural density \(0\). Partial progress was made by Mangerel [Ma22], who proved that this is true for completely additive \(f\) if \[\lvert A\cap [1,X]\rvert \ll \frac{X}{(\log X)^{2+c}}\] for some \(c>0\), and if \(f(p)\) does not have very large values (in a certain technical sense). See also `erdos_491`.