Problem 726

Erdős Problem #726: declared status 'open'. Formalized: no. As $n\to \infty$ ranges over integers$$\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}.$$ Current best: For comparison the classical estimate of Mertens states that$$\sum_{p\leq n}\frac{1}{p}\sim \log\log n.$$By $n\in (p/2,p)\pmod{p}$ we mean $n\equiv r\pmod{p}$ for some integer $r$ with $p/2<r<p$. Prize: no. Tags: number theory.