OPEN
This is open, and cannot be resolved with a finite computation.
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 $p/2<r<p$.
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #726, https://www.erdosproblems.com/726, accessed 2026-01-16
