Truncations of the ring of number-theoretic functions, revisited

Abstract

Let $K$ be a field containing $\mathbb{Q}$, let $Γ$ be the ring of all functions from the positive integers to $K$ under Dirichlet convolution, and let $Γ_n$ be its truncation to functions supported on $[1,n]$. In [Snellman, Homology Homotopy Appl. 2 (2000), 17-27; arXiv:math/9904143] it was shown that $Γ_n$ is a polynomial ring modulo a monomial ideal $I_n$ which is stable after reversing the order of the variables, and the Poincare-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 $\le p_v$. 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 the 2000 paper, which was left open there; the average order $C_n \sim π(n)^2/2$ of the total number of minimal generators, showing that the lower bound $C_n \ge \binom{π(n)+1}{2}$ of that paper is asymptotically sharp, together with the exact order $C_n - \binom{π(n)+1}{2} \sim \frac{16}{3} n^{3/2} / \log^3 n$ of the error; and the identification of $C_n$ with the OEIS sequence A182843. We also record errata for the 2000 paper: one stated result is false, and two proofs are incomplete. Corrected statements and complete proofs are given.