Truncations of the ring of number-theoretic functions, revisited

<jats:p>Let K be a field containing Q, let Γ be the ring of all functions N+ → K under Dirichlet convolution, and let Γn be its truncation to functions supported on [1, n]. In [9] it was shown that Γn 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 Γn was computed in terms of the numbers Cn,v of minimal generators of In of least support v.We prove that Cn,v = Φ(n, pv), Legendre’s sifting function: the number of integers in [1, n] free of prime factors ≤ pv. 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 Cn ∼ π(n)2/2 of the total number of minimal generators, showing that the lower bound Cn ≥ of [9] is asymptotically sharp, together with the exact order  Cn −  ∼ (16/3) n3/2 / log3 n of the error; and the identification of Cn 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.</jats:p>