The density of long runs of non-$r$-free integers

RESUMO

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 $\log(1/D_k)=\frac{r-1}{\zeta(r)}k\log k+\frac{r}{\zeta(r)}k\log\log k-C_rk+o(k)$, where $C_r=\frac{1}{\zeta(r)}((r-1)\log\zeta(r)+r-1+r\log\frac{\pi}{r\sin(\pi/r)})$; for $r=2$ this is $C_2=\frac{6}{\pi^2}(1+\log\frac{\pi^4}{24})=1.4595513\dots$. This determines Mirsky's constant $\alpha(h)$, 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 $\log\alpha(h)\le-\frac54 h\log\log h+O(h)$; 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 $(1/\zeta(r)+o(1))k$ 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 $m\sim k/\zeta(r)$, 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 $\frac{\zeta(r)}{r-1}\frac{\log x}{\log\log x}(1-\frac{(1+o(1))\log\log\log x}{(r-1)\log\log x})$; for $r=2$ the constant is $\pi^2/6$, that of Erdos's 1951 lower bound. We also prove that every covering system of length $k$ has modulus at least $\exp((\frac{r}{\zeta(r)}+o(1))k\log k)$, so the Chinese remainder construction yields only $\zeta(r)/r$, half of $\pi^2/6$ 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.