发表机构
Linköpings Universitet（林雪平大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究重新探讨数论函数环的截断问题，证明截断环的单项式理想极小生成元计数等于勒让德筛选函数，解决了过往猜想，得到生成元总数的渐近阶与误差项，并修正了旧论文的错误与疏漏。
AI 中文摘要
设$K$为包含有理数域$\u211a$的域，$\u0393$为所有从正整数到$K$的函数在狄利克雷卷积下构成的环，$\u0393_n$为其截断，即支撑集在$[1,n]$上的函数构成的环。在[Snellman，《Homology Homotopy Appl.》2（2000），17-27；arXiv:math/9904143]中已证明，$\u0393_n$是一个多项式环模掉一个单项式理想$I_n$，该理想在反转变量顺序后是稳定的，并且$\u0393_n$的庞加莱-贝蒂级数通过$I_n$的最小支撑为$v$的极小生成元个数$C_{n,v}$来计算。我们证明了$C_{n,v} = \u03a6(n,p_v)$，即勒让德筛选函数：$[1,n]$中不含小于等于$p_v$的素因子的整数个数。这将极小自由分解的一个不变量与筛法理论中的经典对象联系起来。作为推论，我们得到：证明了2000年那篇论文中遗留的猜想4.6；极小生成元总数的平均阶为$C_n \u223c \u03c0(n)^2/2$，表明该论文给出的下界$C_n \u2265 \binom{\u03c0(n)+1}{2}$是渐近紧的，同时得到误差的精确阶为$C_n - \binom{\u03c0(n)+1}{2} \u223c \frac{16}{3} n^{3/2} / \u2606^3 n$；并确认$C_n$对应OEIS序列A182843。我们还记录了2000年论文的勘误：一个所述结果是错误的，两个证明是不完整的，并给出了修正后的陈述和完整证明。
英文摘要
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.
Comments23 pages, 2 figures. 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
