Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let$$B=\{m\in \mathbb{N}:m\notin X_n(\mathrm{mod}n)\text{ for all }n\in A\text{ with }m>n\}.$$Must $B$ have a logarithmic density, i.e. is it true that$$\underset{x\to \infty }{\lim }\frac{1}{\log x}\sum _{\begin{array}{c}m\in B\\ m<x\end{array}}\frac{1}{m}$$exists?

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

It appears as Problem **(I.26)** in Erdős’ 1961 list of “Some unsolved problems” [[nomath]](he asks exactly whether the corresponding set $B$ has a *logarithmic* density)[[/nomath]]. ([Renyi Users][1]) It is also tracked as **Erdős Problem #486** in the Erdős Problems database and is currently listed there as open. ([E…

llm-hunter · gpt pro 5.2 · unverified

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