具有小素因子的长整数序列与 $n!$ 的除数函数
Long runs of integers with small prime factors and the divisor function of $n!$
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中文总结 AI 辅助
本文改进 Erdős 等人关于除数函数在阶乘上的增长下界，通过加权 Erdős–Rankin 构造和平均化论证，将下界提升至与 Rankin 素数间隙下界同阶。
中文摘要 AI 辅助
设 $d$ 为除数函数，并设 $K(n)$ 为满足 $d((n + K)!) \ge 2d(n!)$ 的最小正整数 $K$。Erdős、Graham、Ivić 和 Pomerance 证明了，对无穷多个 $n$，\begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} 我们将此结果改进了一个数量级为 $\log_{3} n$ 的因子，这使得该下界与 Rankin 在 1938 年给出的连续素数间隙下界达到同一数量级。这两个问题密切相关，但一个长的无素数区间本身并不能产生大的 $K(n)$ 值：所需的是 Erdős--Rankin 构造的一个加权变体。我们遵循 Erdős、Graham、Ivić 和 Pomerance 的方法，用一个平均化论证替换了一个关键估计，该论证允许某些整数保持未被覆盖。这一改进是在与 Claude Fable 5.1（一个公开可用的生成式人工智能系统）的私下互动中提出并证明的；该论证由作者在此验证并呈现。
英文摘要
Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erdős, Graham, Ivić and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a factor of order $\log_{3} n$, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of $K(n)$: what is needed is a weighted variant of the Erdős--Rankin construction. We follow the method of Erdős, Graham, Ivić and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.
