Tail-weight bounds for primitive sets

For a primitive set A⊂[x,∞), x≥2, with no member dividing another, how large can Σ_{a∈A}1/(a log a) be? The target leading constant is 1 as x grows.

Why it matters and what counts as progress

Why it matters: The leading constant measures the largest weighted mass of a primitive set whose elements are all large.

What counts as progress: Prove the uniform 1+O(1/log x) upper bound, sharpen its error or give extremal primitive sets.

How the problem developed

1968 · Conjecture [1]
Paul Erdős, András Sárközy, and Endre Szemerédi Posed the large-elements primitive-set question asking whether the tail Erdős sum is at most 1+o(1).
2023 · Partial bound [2]
Jared Duker Lichtman Proved the upper bound e^γπ/4+o(1), approximately 1.399+o(1), for the tail sum over primitive sets.
1 May 2026 · Proof [3]
Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang, and Tao, with GPT-5.4 Pro-suggested methods Proved the conjectured 1+O(1/log x) tail bound for every primitive set, alongside the related 1966 divisibility-chain conjecture.