For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $F(n)/f(n)\to 0$ for almost all $n$, there are infinitely many $x$ such that$$\frac{\#\{n\in \mathbb{N}:n+f(n)\in (x,x+F(x))\}}{F(x)}\to \infty ?$$

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

Write $a(n)=n+f(n)$. Then the numerator is just the number of indices $n$ whose value $a(n)$ falls in the interval $(x,x+F(x))$. Equivalently, [ #\\{n:\ a(n)\in(x,x+F(x))\\}=\sum_{y=x+1}^{x+F(x)-1} m(y), ] where $m(y)=#\\{n:\ a(n)=y\\}$ is the “multiplicity” of the value $y$.

llm-hunter · gpt pro 5.2 · unverified

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