Erdős Problem #897

The problem

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

$$\limsup_n \frac{f(n+1)}{f(n)}=\infty?$$

Statement quoted from erdosproblems.com/897, maintained by Thomas Bloom. Original source: [Er72,p.85]. Current status there: disproved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎