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 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].

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