OPEN
As $n\to \infty$ ranges over integers
\[\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}.\]
A conjecture of Erdős, Graham, Ruzsa, and Straus
[EGRS75]. 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 $p/2<r<p$.
