Problem #968: Let $u_n=p_n/n$, where $p_n$ is the $n$th prime

---

Progress

Erdős and Prachar [ErPr61] proved that\[\sum_{p_n<x} \lvert u_{n+1}-u_n\rvert \asymp (\log x)^2,\]and that the set of $n$ such that $u_n>u_{n+1}$ has positive density.

Erdős also asks whether\[u_n<u_{n+1}<u_{n+2}\]or\[u_n>u_{n+1}>u_{n+2}\]have infinitely many solutions.