New upper bound for lattice covering by spheres

We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^{\beta} n \big)$ as $n\to\infty$, where \begin{align*} \beta := \frac{1}{2} \log_2 \left(\frac{8 \pi \mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^{\alpha} n \big)$, where $\alpha := \frac{1}{2} \log_{2}(2\pi \mathrm{e})=2.0471...\,.$