Erdős Problem #858

The problem

Let $A\subseteq \{1,\ldots,N\}$ be such that there is no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ is $>a$. Estimate the maximum of

$$\frac{1}{\log N}\sum_{n\in A}\frac{1}{n}.$$

Statement quoted from erdosproblems.com/858, maintained by Thomas Bloom. Original source: [Er70,p.128]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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