On the critical exponent for $k$-primitive sets

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum 1\mathrm{/}(n\log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$overaprimitivesetismaximizedbytheprimesifandonlyif$\lambda$isatleastthecriticalexponent$\tau_1 \approx 1.14$. A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. One may similarly consider the critical exponent ${\tau }_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that ${\tau }_2<0.8$, which directly implies the Erd\H{o}s conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that ${\tau }_k$ tends to zero as $k\to \mathrm{\infty }$.