Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let$$A=\{n\ge 1:f(n+1)<f(n)\}.$$If $|A\cap [1,X]|=o(X)$ then must $f(n)=c\log n$ for some $c\in \mathbb{R}$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

No — this is not known in full generality.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.