Counting generators of truncation ideals via sieve functions

In the polynomial-ring model of truncated Dirichlet convolution, how many minimal generators of the truncation ideal I_n have least support index v? The proposed identity is Cₙ,ᵥ=Φ(n,pᵥ), where Φ is Legendre’s sifting function.

Why it matters and what counts as progress

Why it matters: The identity links a homological generator count to a classical sieve quantity and informs the earlier Poincaré–Betti conjecture.

What counts as progress: A proof for n≥2 and 1≤v≤π(n), or a counterexample; consequences for the reduced Poincaré–Betti series need their own derivation.

How the problem developed

2000 · Theorem and open conjecture [1]
Jan Snellman Represented the truncated Dirichlet-convolution ring as a polynomial ring modulo a monomial ideal and related its Poincaré–Betti series to minimal-generator counts, leaving Conjecture 4.6 open.
30 Aug 2026 · Theorem [2]
Jan Snellman Proved that the least-support counts of minimal ideal generators equal Legendre’s sifting function, resolving Conjecture 4.6 and deriving total-count asymptotics.

AI contribution

Claude helps identify a sieve-theory formula for truncation ideals

For n ≥ 2 and 1 ≤ v ≤ π(n), the count C_{n,v} of minimal generators of the truncation ideal I_n with least support index v equals Legendre's sifting function Φ(n,p_v).

AI involvementAI-centered

ImportanceModest

Technical verificationPartial

Human understandingFull