Problem Statement
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 $c\in \mathbb{R}$?
Categories:
Number Theory
Progress
Erdős proved that $f(n)=c\log n$ for some $c\in\mathbb{R}$ if $A$ is empty, or if $f(n+1)-f(n)=o(1)$.Partial progress was made by Mangerel [Ma22], who proved that this is true if\[\lvert A\cap [1,X]\rvert \ll \frac{X}{(\log X)^{2+c}}\]for some $c>0$, and if $g(p)$ does not have very large values (in a certain technical sense).
See also [491].
Source: erdosproblems.com/1122 | Last verified: January 19, 2026