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

Abstract

Fix $r\ge 2$ 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 $\log (1/D_k)=\frac{r-1}{\zeta (r)}k\log k+\frac{r}{\zeta (r)}k\log \log k-C_{r}k+o(k)$, 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 $\log \alpha (h)\le -\frac{5}{4}h\log \log h+O(h)$; 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.

arXiv comment15 pages, 3 figures, 6 tables