Problem 897

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?

Source: erdosproblems.com/897

Number theory

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

The site labels the problem DISPROVED (LEAN) (page last edited 1 April 2026). The counterexample is recorded on the claim page Wirsing 1981; its 2025 rediscovery and the Lean formalization of that rediscovery on Archivara 2025.

Problem 897

Status

Proof claims

Discussion