  We show there exists a packing of identical spheres in $\mathbb{R}^d$ with
density at least
  \[
  (1-o(1))\frac{d \log d}{2^{d+1}}\, ,
  \]
  as $d\to\infty$. This improves upon previous bounds for general $d$ by a
factor of order $\log d$ and is the first asymptotically growing improvement to
Rogers' bound from 1947.
