  In [K. Bou-Rabee, B. Seward, J. Reine Angwe. Math. 2016] Bou-Rabee and Seward
constructed examples of finitely generated residually finite groups $G$ whose
residual finiteness growth function $\mathcal{F}_G$ can be at least as fast as
any prescribed function. In this note we describe a modified version of their
construction, which allows us to give a complementary upper bound on
$\mathcal{F}_G$. As such, every nondecreasing function at least $\exp ( n \log
(n)^2 \log \log (n)^{1+\epsilon} )$ is close to the residual finiteness growth
function of some finitely generated group. We also have similar result for the
full residual finiteness growth function and for the divisibility function.
