Let K be a field containing Q, let Γ be the ring of all functions N⁺ → K under Dirichlet convolution, and let Γ_n be its truncation to functions supported on [1, n]. In [9] it was shown that Γ_n is a polynomial ring modulo a monomial ideal In which is stable after reversing the order of the variables, and the Poincaré–Betti series of Γ_n was computed in terms of the numbers C_n,v of minimal generators of I_n of least support v.

We prove that C_n,v = Φ(n, p_v), Legendre’s sifting function: the number of integers in [1, n] free of prime factors ≤ p_v. This identifies an invariant of a minimal free resolution with a classical object of sieve theory. As consequences we obtain: a proof of Conjecture 4.6 of [9], which was left open there; the average order C_n ∼ π(n)²/2 of the total number of minimal generators, showing that the lower bound C_n ≥ of [9] is asymptotically sharp, together with the exact order C_n − ∼ (16/3) n^3/2 / log³n of the error; and the identification of C_n with the OEIS sequence A182843. We also record errata for [9]: one stated result is false, and two proofs are incomplete. Corrected statements and complete proofs are given.