OPEN
This is open, and cannot be resolved with a finite computation.
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 $c\in \mathbb{R}$?
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].
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This page was last edited 30 December 2025.
Additional thanks to: Alfaiz
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #1122, https://www.erdosproblems.com/1122, accessed 2026-01-16
