Problem Statement
Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). 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\[f(n)=c'\log n+O(1)?\]
Categories:
Number Theory
Progress
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].
Source: erdosproblems.com/491 | Last verified: January 15, 2026