The density of long runs of non-r-free integers
Abstract
Fix r≥2 and let Dk be the natural density of the n such that none of n+1,…,n+k is r-free. We prove that log(1/Dk)=(r−1)/ζ(r)klog k+r/ζ(r)kloglog k−Crk+o(k), where Cr=1/ζ(r)((r−1)logζ(r)+r−1+rlogπ/rsin(π/r)); for r=2 this is C2=6/π2(1+logπ4/24)=1.4595513…. This determines Mirsky's constant α(h), the density of the n with sn+1−sn=h in the squarefree numbers, to the same precision. Previously the only known estimate was logα(h)≤−5/4 hloglog h+O(h); ours improves this by a factor of order log h/loglog h, and is two-sided. The upper bound rests on the observation that the primes p≤ z with ∏p≤ zpr≤√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 k1/r. The resulting sum over injections is m!em with m∼ k/ζ(r), evaluated by a saddle point. For the lower bound the Chung-Erdos inequality recovers the full contribution of m!em 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 ζ(r)/(r−1)(log x)/(loglog x)(1−((1+o(1))logloglog x)/((r−1)loglog x)); for r=2 the constant is π2/6, that of Erdos's 1951 lower bound. We also prove that every covering system of length k has modulus at least exp((r/ζ(r)+o(1))klog k), so the Chinese remainder construction yields only ζ(r)/r, half of π2/6 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.
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BibTeX
@misc{jiang2026density,
title = {{The density of long runs of non-$r$-free integers}},
author = {Xinyang Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09221},
eprint = {2609.09221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09221},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The density of long runs of non-r-free integers
AU - Xinyang Jiang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.09221
UR - https://arxiv.org/abs/2609.09221
AB - Fix r≥2 and let D_k be the natural density of the n such that none of n+1,…,n+k is r-free. We prove that log(1/D_k)=(r−1)/ζ(r)klog k+r/ζ(r)kloglog k−C_rk+o(k), where C_r=1/ζ(r)((r−1)logζ(r)+r−1+rlogπ/rsin(π/r)); for r=2 this is C_2=6/π^2(1+logπ^4/24)=1.4595513…. 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)≤−5/4 hloglog h+O(h); ours improves this by a factor of order log h/loglog h, and is two-sided. The upper bound rests on the observation that the primes p≤ z with ∏_p≤ zp^r≤√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∼ 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 ζ(r)/(r−1)(log x)/(loglog x)(1−((1+o(1))logloglog x)/((r−1)loglog 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((r/ζ(r)+o(1))klog 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.
ER -