What was already known
Snellman's 2000 paper expressed C_{n,v} as a count of rough integers in (n/p_v,n], but had not identified it with Legendre's sifting function. Its Conjecture 4.6 about the reduced Poincaré–Betti series remained open.
AI involvement
The author explicitly credits Claude with finding Theorem 4, its proof, and its graded refinement in interactive sessions. Jan Snellman supplied the original 2000 problem, directed the exposition, checked results, and takes responsibility for the paper.
Snellman explicitly credits Claude with finding Theorem 4, its proof, and its graded refinement in interactive sessions. Snellman supplied the research context, directed the work and checked the result.
- Supporting source (arxiv.org) — Theorem 4 and Disclosure of AI assistance
Importance
The identity connects a minimal free resolution invariant to a classical sieve count and supplies a short proof route to the earlier conjecture.
Limitations
The displayed identity is an all-n theorem in the manuscript. Its Lean files do not formalize the main identity or the truncation ideal: they define C_{n,v} using Φ and check sample values. The author reports independent SageMath numerical checks and personal review; these do not certify the general theorem. The paper's graded refinement and asymptotic claims are related results, not independently assessed here.
Technical verification
The author reports independent numerical checks of the generator counts and personally reviewing results. The paper expressly says the general identity is not formalized in Lean: the Lean code takes it as a definition and checks examples. No completed rigorous check of the general claim is documented.
Human understanding
Snellman explains the central telescoping argument in the proof of Theorem 4 and discusses why the algebraic generator count becomes a sieve count. His account shows human understanding of this scoped identity.
Correction and publication history
Primary manuscript Theorem 4, prior 2000 baseline, explicit AI attribution, and formalization limits reviewed; non-notifying local inclusion.
Read this published revision