Truncations of the ring of number-theoretic functions, revisited
Abstract
Let K be a field containing ℚ, let Γ be the ring of all functions from the positive integers to K under Dirichlet convolution, and let Γn 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 Γn is a polynomial ring modulo a monomial ideal In which is stable after reversing the order of the variables, and the Poincare-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, 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 Cn ∼ π(n)2/2 of the total number of minimal generators, showing that the lower bound Cn ≥ binomπ(n)+12 of that paper is asymptotically sharp, together with the exact order Cn − binomπ(n)+12 ∼ 16/3 n3/2 / log3 n of the error; and the identification of Cn 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.
Versions
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BibTeX
@misc{snellman2026truncations,
title = {{Truncations of the ring of number-theoretic functions, revisited}},
author = {Jan Snellman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29555},
eprint = {2608.29555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29555},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Truncations of the ring of number-theoretic functions, revisited
AU - Jan Snellman
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.29555
UR - https://arxiv.org/abs/2608.29555
AB - Let K be a field containing 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 ≤ 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 ∼ π(n)^2/2 of the total number of minimal generators, showing that the lower bound C_n ≥ binomπ(n)+12 of that paper is asymptotically sharp, together with the exact order C_n − binomπ(n)+12 ∼ 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.
ER -