The Product-Divisor Set Problem: A Multiplicative Variant of Erdős's Primitive Set Problem

We introduce a strengthening of the classical primitive set condition. Let A be a subset of the positive integers such that no distinct a, b, c ∈ A satisfy a | bc. We determine the asymptotic behavior of the maximal reciprocal sum S(x) = ∑_{a ∈ A, a ≤ x} 1/a. We prove that the maximum is achieved by the set of prime numbers, yielding S(x) ∼ log log x. This contrasts with classical primitive sets, where divisibility is only forbidden between pairs (a ∤ b), allowing for significantly denser sets with sum ∼ log x / √(log log x). This is a collaborative work between human author Britt B. and Grok (xAI), exploring a new number theory problem inspired by Erdős.