New upper bound for lattice covering by spheres

We show that there exists a lattice covering of Rⁿ by Eucledian spheres of equal radius with density O(n ln^β n ) as n→∞, where {align*} β:= 1/2 log_2 (8 πe/3√ 3)=1.85837...\,. {align*} This improves upon the previously best known upper bound by Rogers from 1959 of O(n ln^α n ), where α:= 1/2 log₂(2πe)=2.0471...\,.