OPEN
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
\[\frac{1}{\log N}\sum_{n\in A}\frac{1}{n}.\]
Alexander
[Al66] and Erdős, Sárközi, and Szemerédi
[ESS68] proved that this maximum is $o(1)$ (as $N\to \infty$). This condition on $A$ is a weaker form of the usual primitive condition. If $A$ is merely primitive then Behrend
[Be35] proved
\[\frac{1}{\log N}\sum_{n\in A}\frac{1}{n}\ll \frac{1}{\sqrt{\log\log N}}.\]
An example of such a set $A$ is the set of all integers in $[N^{1/2},N]$ divisible by some prime $>N^{1/2}$.
See also [143].
