Problem 486

Erdős Problem #486: declared status 'open'. Formalized: yes. Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let$$B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\textrm{ with }m>n\}.$$Must $B$ have a logarithmic density, i.e. is it true that$$\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}$$exists? Prize: no. Tags: number theory, primitive sets.