Truncations of the ring of number-theoretic functions, revisited

Abstract

Let $K$ be a field containing $\mathbb{Q}$, let $\Gamma$ be the ring of all functions from the positive integers to $K$ under Dirichlet convolution, and let ${\Gamma }_n$ 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 ${\Gamma }_n$ 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 ${\Gamma }_n$ was computed in terms of the numbers $C_{n,v}$ of minimal generators of $I_n$ of least support $v$. We prove that $C_{n,v}=\Phi (n,p_v)$, 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, and it refines to the grading by total degree, with the number of prime factors carried in a variable. As consequences we obtain: a proof of Conjecture 4.6 of the 2000 paper, which was left open there; the asymptotics $C_n\sim \pi (n{)}^2/2$ of the total number of minimal generators, showing that the lower bound $C_n\ge \left(\genfrac{}{}{0ex}{}{\pi (n)+1}{2}\right)$ of that paper is asymptotically sharp, together with the exact order $C_n-\left(\genfrac{}{}{0ex}{}{\pi (n)+1}{2}\right)\sim \frac{16}{3}{n}^{3/2}/{\log }^{3}n$ 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, one worked example is miscomputed, and two proofs are incomplete. Corrected statements and complete proofs are given.

arXiv comment25 pages, 2 figures. v2: adds Theorem 4', the graded refinement of the main theorem, and Section 4.2, an erratum to Example 4.5 of the 2000 paper. Written with substantial assistance from a large language model (Claude, Anthropic); see the "Disclosure of AI assistance" section. The author takes full responsibility for the results