OPEN
Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that
\[\limsup_{p,k}\frac{f(p^k)}{\log p^k}=\infty.\]
Is it true that
\[\limsup_n \frac{f(n+1)-f(n)}{\log n}=\infty?\]
Or perhaps even
\[\limsup_n \frac{f(n+1)}{f(n)}=\infty?\]
A conjecture of Erdős and Wirsing. Wirsing
[Wi70] proved (see
[491]) that if $\lvert f(n+1)-f(n)\rvert \leq C$ then $f(n)=c\log n+O(1)$ for some constant $c$.
Erdős suggests that for simplicity one might first assume that $f(p^k)=f(p)$ or $f(p^k)=kf(p)$.
