AI 中文总结
本文研究长段非$r$-自由整数序列的密度，通过筛法和鞍点法精确确定其渐近对数，并改进Mirsky常数，证明Erdos下界等价于第一矩阈值。
AI 中文摘要
固定$r\ge2$，设$D_k$为使得$n+1,\dots,n+k$中没有一个数是$r$-自由的$n$的自然密度。我们证明$\log(1/D_k)=\frac{r-1}{\zeta(r)}k\log k+\frac{r}{\zeta(r)}k\log\log k-C_rk+o(k)$，其中$C_r=\frac{1}{\zeta(r)}((r-1)\log\zeta(r)+r-1+r\log\frac{\pi}{r\sin(\pi/r)})$；对于$r=2$，$C_2=\frac{6}{\pi^2}(1+\log\frac{\pi^4}{24})=1.4595513\dots$。这以相同的精度确定了Mirsky常数$\alpha(h)$，即平方自由数中满足$s_{n+1}-s_n=h$的$n$的密度。此前唯一的已知估计是$\log\alpha(h)\le-\frac54 h\log\log h+O(h)$；我们的结果将其改进了约$\log h/\log\log h$倍，并且是双侧的。上界依赖于观察：满足$\prod_{p\le z}p^r\le\sqrt k$的素数$p\le z$筛除长度为$k$的窗口时没有对齐的自由度，因此至少有$(1/\zeta(r)+o(1))k$个位置必须被大于$k^{1/r}$的不同素数的$r$次幂覆盖。由此得到的对注入的求和为$m!e_m$，其中$m\sim k/\zeta(r)$，通过鞍点法求值。对于下界，Chung-Erdos不等式恢复了$m!e_m$的全部贡献，而不仅仅是其最大项，这正是线性系数显式化的原因。作为推论，$r$-自由数之间间隙的第一矩阈值是$\frac{\zeta(r)}{r-1}\frac{\log x}{\log\log x}(1-\frac{(1+o(1))\log\log\log x}{(r-1)\log\log x})$；对于$r=2$，常数为$\pi^2/6$，即Erdos 1951年下界中的常数。我们还证明了每个长度为$k$的覆盖系统的模至少为$\exp((\frac{r}{\zeta(r)}+o(1))k\log k)$，因此中国剩余定理构造仅给出$\zeta(r)/r$，当$r=2$时为$\pi^2/6$的一半。因此，Erdos的界等价于断言：$k$个连续非$r$-自由整数的首个游程出现在第一矩阈值处，而非由覆盖系统的最小模设定的尺度。
英文摘要
Fix $r\ge2$ and let $D_k$ be the natural density of the $n$ such that none of $n+1,\dots,n+k$ is $r$-free. We prove that $\log(1/D_k)=\frac{r-1}{ζ(r)}k\log k+\frac{r}{ζ(r)}k\log\log k-C_rk+o(k)$, where $C_r=\frac{1}{ζ(r)}((r-1)\logζ(r)+r-1+r\log\fracπ{r\sin(π/r)})$; for $r=2$ this is $C_2=\frac{6}{π^2}(1+\log\frac{π^4}{24})=1.4595513\dots$. This determines Mirsky's constant $α(h)$, the density of the $n$ with $s_{n+1}-s_n=h$ in the squarefree numbers, to the same precision. Previously the only known estimate was $\logα(h)\le-\frac54 h\log\log h+O(h)$; ours improves this by a factor of order $\log h/\log\log h$, and is two-sided. The upper bound rests on the observation that the primes $p\le z$ with $\prod_{p\le z}p^r\le\sqrt k$ sieve a window of length $k$ with no freedom of alignment, so that at least $(1/ζ(r)+o(1))k$ positions must be covered by $r$-th powers of distinct primes exceeding $k^{1/r}$. The resulting sum over injections is $m!e_m$ with $m\sim k/ζ(r)$, evaluated by a saddle point. For the lower bound the Chung-Erdos inequality recovers the full contribution of $m!e_m$ rather than merely its largest term, which is what makes the linear coefficient explicit. As a consequence the first-moment threshold for gaps between $r$-free numbers is $\frac{ζ(r)}{r-1}\frac{\log x}{\log\log x}(1-\frac{(1+o(1))\log\log\log x}{(r-1)\log\log x})$; for $r=2$ the constant is $π^2/6$, that of Erdos's 1951 lower bound. We also prove that every covering system of length $k$ has modulus at least $\exp((\frac{r}{ζ(r)}+o(1))k\log k)$, so the Chinese remainder construction yields only $ζ(r)/r$, half of $π^2/6$ when $r=2$. Erdos's bound is thus equivalent to the assertion that the first run of $k$ consecutive non-$r$-free integers occurs at the first-moment threshold, not at the scale set by the minimal modulus of a covering system.
Comments15 pages, 3 figures, 6 tables
