PROVED
This has been solved in the affirmative.
Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). If there is a constant $c$ such that $\lvert f(n+1)-f(n)\rvert <c$ for all $n$ then must there exist some $c'$ such that\[f(n)=c'\log n+O(1)?\]
Erdős
[Er46] proved that if $f(n+1)-f(n)=o(1)$ or $f(n+1)\geq f(n)$ then $f(n)=c\log n$ for some constant $c$.
This is true, and was proved by Wirsing
[Wi70].
See also
[897] and
[1122].
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This page was last edited 30 December 2025.
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