Banks–Martin weighted inequalities for primitive sets
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For a primitive set A of integers, with no member dividing another and each member having at least k prime factors, let A(Q) contain those built from Q and let Nₖ(Q) contain numbers with exactly k prime factors from Q, counted with multiplicity. Is Σ_{a∈A(Q)}1/(a log a) at most the corresponding sum over Nₖ(Q), for every k≥1 and every set Q of odd primes?
Why it matters and what counts as progress
Why it matters: The odd-prime restriction repairs a false unrestricted primitive-set inequality while preserving its central weighted extremal question.
What counts as progress: Prove the inequality for all odd-prime sets or extend known parameter classes. Allowing the prime 2 gives a different, false unrestricted claim.
How the problem developed
- 2013 · Conjecture [1]
William D. Banks and Greg Martin Proposed monotonicity of the weighted sums over integers with k prime factors and broader inequalities for primitive sets supported on prime subsets.
- 2019 · Counterexample [2]
Jared Duker Lichtman Disproved the unrestricted monotonicity conjecture: the sum over k-almost-primes has a global minimum at k=6.
- 1 May 2026 · Reported claim [3]
Alexeev and coauthors, with a method suggested by GPT-5.4 Pro Proved a revised Banks–Martin inequality when the allowed prime set consists of odd primes; the original form allowing all primes is not recovered.
AI contribution
Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
For k≥1, a primitive set A of integers with at least k prime factors, and any set Q of odd primes, the sum f(A(Q)) is at most f(Nₖ(Q)), where f sums 1/(a log a).
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